NORMAL SETS IN AMENABLE SEMIGROUPS
IN MEMORY OF ROY ADLER
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
Abstract. Let (Fn ) be a (left) Følner sequence in a countable amenable
semigroup G which is embeddable in a group. We introduce the notion of
(Fn )-normal set in G and (Fn )-normal sequence in {0, 1}G . When G = (N, +)
and Fn = {1, 2, ..., n}, the (Fn )-normality coincides with the classical notion.
We prove several results about this notion, for example:
• If (Fn ) is a Følner sequence in a countable amenable group G, such that
P −|F |
n < ∞ then almost every x ∈ {0, 1}G is (F )-normal.
n
ne
• In any amenable semigroup which is embeddable in a group, given any
Følner sequence (Fn ), there exists a Champernowne-like effective (Fn )normal set.
We also investigate and juxtapose combinatorial and Diophantine properties
of normal sets in semigroups (N, +) and (N, ×). Below is a sample of results
that we obtain:
• Let A ⊂ N be a classical normal set. Then, for any Følner sequence (Kn )
in (N, ×) there exists a set E of (Kn )-density 1, such that for any finite
subset {n1 , n2 , . . . , nk } ⊂ E, the intersection A/n1 ∩A/n2 ∩. . .∩A/nk has
positive upper density in (N, +). As a consequence, A contains arbitrarily
long geometric progressions, and, more generally, arbitrarily long “geoarithmetic” configurations of the form {a(b + ic)j , 0 ≤ i, j ≤ k}.
• There is a rather natural and sufficiently large class of Følner sequences
(Fn ) in (N, ×), which we call “nice”, and which have the property that
there exists a Liouville number, whose binary expansion is (Fn )-normal.
• In (N, ×) there exists a Champernowne-like set which is (Fn )-normal for
every nice Følner sequence (Fn ).
• Any set which is normal with respect to every nice Følner sequence contains, for each k ∈ N, solutions a, b, c of the equation ab = ck .
1. Introduction
It follows from the classical law of large numbers ([Bo]) that given a fair coin
whose sides are labeled 0 and 1, the infinite binary sequence (xn ) obtained by
independent tossing of the coin, is almost surely normal, meaning that, for any
k ∈ N = {1, 2, . . . }, any 0-1 word of length k appears in (xn ) with frequency 1/2k .
This provides a proof of existence of normal sequences (note that a priori it is
not even clear whether normal sequences exist!). There are also numerous explicit
constructions of normal sequences (see for example [Ch, vM, DE]). For example, the
Champernowne sequence 1 10 11 100 101 110 . . . , which is formed by the sequence
1, 2, 3, 4, 5, 6, . . . written in base 2, is a normal sequence.
Any 0-1 sequence
(xn ) ∈ {0, 1}N may be viewed as the sequence of digits in the
P∞
binary expansion n=1 xn /2n ∈ [0, 1], which leads to an equivalent formulation
of the above fact: almost every x ∈ [0, 1] is normal in base 2. Similarly, due to
Date: June 4, 2017.
1
2
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
the natural bijection between 0-1 sequences and subsets of N (any subset S ⊂ N is
identified with the sequence (xn ), where xn = 1 if and only if n ∈ S), one can talk
about normal sets in N (more accurately, in (N, +); see the discussion below).
The peculiar combinatorial and Diophantine properties of normal sets/numbers,
together with the fact that they are “typical,” make them a natural object of interest
and a source of various generalizations, see [AKS, Ko, PV, BB, Fi1]. In this paper
we introduce the notion of a normal set for any countable amenable group (indeed,
for any countable amenable semigroup which embeds in a group). In order to do
so, one has to give meaning to the statement
• A sequence (xg )g∈G ∈ {0, 1}G is normal if any finite 0-1 “word” of “length”
k “occurs” in (xg )g∈G with the “frequency” 2−k .
We start by generalizing the notion of a word. It will be convenient to identify
subsets A ⊂ G with their indicator functions, which are elements of {0, 1}G (and
will be denoted by the letter xA ). Let K ⊂ G be a finite set and let w ∈ {0, 1}K
be a 0-1 function defined on K (we will call it a block ). The natural analogue of
the “length” of w is the cardinality |K| of its domain K. Consider the intersection
\
V (w, A) =
k −1 Aw(k) ,
k∈K
where A1 = A and A0 = G \ A, and k −1 A = {g ∈ G : kg ∈ A} (which makes sense
even if k −1 formally does not exist in G).
An element g belongs to V (w, A) if and only if
(1.1)
∀k ∈ K xA (kg) = w(k).
Define the action g 7→ σg of G on the symbolic space {0, 1}G by
(σg (x))(h) = x(hg),
where x = (x(g))g∈G ∈ {0, 1}G . The condition (1.1) says that the image of xA
under the map σg , restricted to K, coincides with the block w. If we denote by
[w] the cylinder set in {0, 1}G corresponding to the block w then we can simply
write σg (xA ) ∈ [w]. So, V (w, A) can be viewed as either the set of occurences of
the (shifted) block w in xA , or, equivalently, as the set of times when xA visits [w].
Let us now observe that the discussed above classical notions of normal sequences
and normal sets depend heavily on the preferred (and in the case of N, quite natural)
ordering, which leads to the averaging procedure that determines the notion of
“frequency”. The “frequency” in question is defined as limn→∞ νn (w)/n, where
νn (w) is the number of occurrences of the 0-1 word w = w1 w2 . . . wk in (xj )nj=1 ,
i.e.,
νn (w) = |{m ∈ {1, . . . , n − k + 1} : xm+j−1 = wj for 1 ≤ j ≤ k}|.
To define “frequency” in the general case we have to use our additional assumption
that the countable semigroup G is cancellative and amenable, and hence has Følner
sequences. A sequence (Fn ) of finite subsets of G is called (left) Følner if for any
g ∈ G one has
lim
n→∞
|gFn ∩ Fn |
= 1.
|Fn |
NORMAL SETS IN AMENABLE SEMIGROUPS
3
Given a Følner sequence (Fn ) of subsets of G and a set V ⊂ G, one defines the
upper and lower (Fn )-densities d(Fn ) (V ) and d(Fn ) (V ) by the formulas
|Fn ∩ V |
,
|Fn |
n→∞
|Fn ∩ V |
d(Fn ) (V ) = lim inf
.
n→∞
|Fn |
d(Fn ) (V ) = lim sup
When d(Fn ) (V ) = d(Fn ) (V ), we say that V has (Fn )-density (which is that common
value) and denote it by d(Fn ) (V ).
We now are in a position to define the main object of our interest, the general
notion of normality, which in view of the above discussion, is a direct extension of
the classical notion.
Definition 1.1. Let G be a cancellative countable amenable semigroup and let (Fn )
be a Følner sequence in G. A set A ⊂ G (and its indicator function xA ∈ {0, 1}G )
is called (Fn )-normal if for any block w (cylinder set [w]) over a finite set K, we
have
d(Fn ) (V (w, A)) = 2−|K| .
The above definition can be stated in the following equivalent form:
Definition 1.2. A set A is (Fn )-normal if, for any finite set K = {k1 , k2 , . . . , kr }
the following holds:
d(Fn ) (k1 A ∩ k2 A ∩ · · · ∩ kr A) = 2−r .
(Note that the equation in Definition 1.2 is the same as that in Definition 1.1 applied
to a block w which consists of only 1’s, so Definition 1.1 implies Definition 1.2.
The converse implication uses the fact that a family of independent sets remains
independent if some of them are replaced by their complements.)
It is obvious from Definition 1.2 that for any g ∈ G we have:
(1) d(Fn ) (V ) = d(Fn ) (gV ) = d(Fn ) (g −1 V ), and
(2) A is (Fn )-normal if and only if so is gA and g −1 A. If the semigroup G is
commutative then x ∈ {0, 1}G is (Fn )-normal if and only if so is σg (x).1
As noted above, the classical normal sets correspond to the Følner sequence
Fn = {1, . . . , n} ⊂ (N, +). But clearly one has uncountably many notions of (Fn )normality in (N, +).
An additional important feature of classical normality is that almost every binary sequence x ∈ {0, 1}N is normal [Bo]. Here “almost every” refers to the product
measure ( 12 , 21 )N , or, when one thinks of binary sequences as representing – via the
binary expansion – the numbers of [0, 1], it refers to the Lebesgue measure. Nowadays this fact is routinely derived from the Birkhoff’s pointwise ergodic theorem.
A general pointwise ergodic theorem for amenable groups, which was obtained by
Lindenstrauss in [Li], implies that if (Fn ) is a so-called tempered Følner sequence,
i.e., satisfies for every n ≥ 1 the condition
n
[
−1
Fi Fn+1 ≤ C|Fn+1 |,
i=1
1See Remark 2.5 for the discussion of the noncommutative case.
4
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
then almost every x ∈ {0, 1}G is (Fn )-normal. However, not every Følner sequence
is tempered, and, moreover, not along every Følner sequence the pointwise ergodic
theorem holds. For example, it is proved in [AdJ] that for any aperiodic measure
preserving system (X, B, µ, T ) there exists an L∞ function f such that the almost
everywhere convergence
of ergodic averages along the specific Følner sequence Fn =
√
{n, n + 1, . . . , n + b nc} does not hold. Nevertheless, as it will be shown in the next
section, without using the Lindenstrauss’ ergodic theorem, the following theorem
holds.
Theorem 1.3 (cf. TheoremP
2.3 below). If (Fn ) is a Følner sequence in a countable
∞
−|Fn |
amenable group, such that
< ∞, then almost every x ∈ {0, 1}G is
n=1 e
(Fn )-normal.
Returning to the subsets of N, it is natural to investigate normality in the cancellative semigroup (N, ×) and juxtapose the properties of the corresponding multiplicatively normal sets with those of additively normal sets. Here are some of the
questions which will be addressed in this paper.
• Which types of combinatorial configurations can always be found in normal
subsets of (N, +) and (N, ×)?
• What are universal combinatorial properties that are shared by (Fn )-normal
sets in (N, +), or in (N, ×), for all Følner sequences?
• Are there Følner sequences in (N, +) or (N, ×) which guarantee some additional combinatorial and Diophantine richness of the corresponding families
of normal sets?
• Are there natural Champernowne-like constructions in (N, ×), and, more
generally, in the set-up of countable amenable semigroups?
• Are there Liouville numbers (viewed via the binary representation) which
are multiplicatively normal? (It is known that there are Liouville numbers
that are normal in the classical sense, that is, ({1, . . . , n})-normal (see [Bu]
and the references therein).
Here is a sample of results obtained in this paper which provide some answers
to these questions.
• Let (Fn ) be any Følner sequence in (N, +). Then any (Fn )-normal set S
contains solutions of any partition regular system of linear equations.2
• Let (Fn ) be any Følner sequence in (N, ×). Then any (Fn )-normal set
S contains solutions of any homogeneous system of polynomial equations
which has solutions in N.
• Let (Fn ) be any Følner sequence in (N, ×). Then any (Fn )-normal set
S contains solutions of any homogeneous system of polynomial equations
which has solutions in N.
• Let S be any classical normal set in (N, +). Then
(i) S contains solutions a, b, c of any equation ia + jb = kc, where i, j, k
are arbitrary positive integers,
(ii) S contains pairs {n + m, nm} with arbitrary large n, m,
(iii) S contains arbitrarily long geometric progressions, and, more generally, arbitrarily long “geo-arithmetic” configurations of the form
{a(b + ic)j , 0 ≤ i, j ≤ k}.
2A system of equations is called partition regular if for any coloring of N there exist a monochromatic solution.
NORMAL SETS IN AMENABLE SEMIGROUPS
5
• In any countable amenable group, and indeed in any amenable semigroup
which embedds in a group, given any Følner sequence (Fn ) (even such along
which the pointwise ergodic theorem fails), there exists a Champernownelike effective (Fn )-normal set.
• There is a rather natural and sufficiently large class of Følner sequences
(Fn ) in (N, ×), which we call “nice”, and which have the property that
there exists an (Fn )-normal Liouville number.
• In (N, ×) there exists a Champernowne-like set which is (Fn )-normal for
every nice Følner sequence (Fn ).
• Any set which is normal with respect to every nice Følner sequence contains,
for each k ∈ N, solutions a, b, c of the equation ab = ck .
2. Normality as a dynamical property
Fix a countable amenable semigroup G and a Følner sequence (Fn ) in G. Suppose
that G acts by continuous maps Tg on a compact metric space X, preserving a
probability measure µ. A point x ∈ X is called (Fn )-generic for µ if for any
continuous function f ∈ C(X) we have the following convergence of the Césaro
averages
Z
1 X
f (Tg x) = f dµ.
(2.1)
lim
n→∞ |Fn |
g∈Fn
It is a standard fact that when X = {0, 1}G and µ is a Borel probability measure
invariant with respect to the shifts (σg )g∈G , then a point x is generic for µ if and
only if (2.1) holds for all indicator functions of cylinder sets in X. Note that
the action (σg )g∈G preserves (among many other measures) the uniform Bernoulli
measure, henceforth denoted by λ, which is simply the product measure mG , with m
denoting the uniform ( 21 , 12 )-measure on {0, 1}. Thus, by an elementary translation
of the above notions we get yet another condition equivalent to normality of a set
A ⊂ G:
Fact 2.1. A set A ⊂ G is (Fn )-normal if and only if its indicator function xA is
(Fn )-generic for the Bernoulli measure λ on {0, 1}G .
As mentioned in the Introduction, derivation of the existence (and full measure)
of the set of (Fn )-normal 0-1 sequences from the pointwise ergodic theorem requires
the Følner sets to be rather special. However, in the specific case of the Bernoulli
measure and cylinder sets, the conventional pointwise ergodic theorem can be replaced by Theorem 2.3 below, which is valid under much weaker restrictions. The
formulation of this theorem will be preceded by some preparatory material needed
in the proof.
We start with a lemma allowing to restrict our attention to groups (rather than
semigroups). The lemma implies that anything we prove for countable amenable
groups applies to all “reasonable” countable amenable semigroups as well.
Lemma 2.2. Let (Fn ) be a Følner sequence in an amenable semigroup S which is
embeddable in a group.3 Then there exists an amenable group G containing S as a
subsemigroup, such that (Fn ) is a Følner sequence in G.
3For instance, S can be cancellative and commutative or cancellative and right reversible, see
[CP]. See [La] for necessary and sufficient conditions.
6
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
Proof. Let us first embed S in some group G0 . Then each element s ∈ S has
a definite inverse s−1 ∈ G0 . Define G ⊂ G0 as the set of all finite products
−1
s1 s−1
2 · · · s2k−1 s2k where all terms si (i = 1, . . . , 2k) belong to S ∪ {e} (e is the
0
unity of G and must be added only in case S does not have a unity). Clearly, G is
a subgroup of G0 and it contains S. Fix an element g ∈ G and write it as a product
−1
ε
0
s1 s−1
2 · · · s2k−1 s2k . Also fix an ε > 0 and denote ε = 2k . For large n, the set Fn
0
is (si , ε )-invariant for each i = 1, . . . , 2k. This implies that at least 1 − ε0 percent
of elements of Fn has the form s2k f with f ∈ Fn . Thus the same percentage of
0
elements of gFn has the form s1 s−1
2 · · · s2k−1 f (f ∈ Fn ). Further, another 1 − ε
0
percent of all elements of s2k−1 Fn belong to Fn (can be written as f ∈ Fn ). That
−1
0
is, at least 1 − 2ε0 percent of elements of gF has the form s1 s−1
2 · · · s2k−3 s2k−2 f
0
0
(f ∈ Fn ). Repeating this (double) argument k times we prove that 1 − 2kε = 1 − ε
percent of all elements of gF belong to F , i.e., that (Fn ) is (g, ε)-invariant. We
have shown that (Fn ) is a Følner sequence in G. This also implies that the group
G is amenable.
Our key tool for handling actions of a countable amenable group G is a special
system of tilings (Tk )k≥1 of G built in [DHZ] (we could manage with an older
concept of quasi-tilings [OW], but the above tilings deliver a much more convenient
tool). Below we explain all involved terms.
A tiling T of G is determined as a pair (S, C), where S is a collection of finite
sets called shapes and C = {CS : S ∈ S} is a collection of (usually infinite) center
sets, such that the family
T = {Sc : S ∈ S, c ∈ CS }
is a partition of G. Without changing the tiling, we can always shift the shapes
S so that they contain the unity, which will result in shifting the center sets so
that each tile contains its center. In particular, the sets CS become disjoint for
different shapes S. An element Sc of this partition will be called a tile of shape S
centered at c. By disjointness of the tiles, the assignment (S, c) 7→ Sc is injective
from {(S, c) : S ∈ S, c ∈ CS } into T , i.e., each tile has a uniquely determined center
and shape.
Given a tiling T and a set F ⊂ G, by the T -saturation of F we will understand
the union of all tiles of T which intersect F . It is an easy observation, that if F is
finite and (K, ε)-invariant,4 where K is the union of the sets SS −1 over all shapes
S of these tiles of T which intersect F , then F differs from its saturation by at
most ε|F | elements.
A tiling whose set of shapes is finite will be called proper.
A sequence of proper tilings Tk is called a congruent system of tilings if for each
k every tile of Tk+1 equals a union of some tiles of Tk . A congruent system of tilings
is deterministic, if any two tiles of Tk+1 of the same shape are partitioned into the
tiles of Tk the same way, up to shifting. In other words, this partition depends on
the shapes, not tiles, of Tk+1 . By iterating, the same applies to partitioning the
shapes of Tk+1 into tiles of Ti for any 1 ≤ i ≤ k. In the deterministic case, the
tiling Tk+1 determines all the tilings T1 , . . . , Tk .
4A set F is (K, ε)-invariant if |KF 4F | < ε. It is easy to see that (F ) is a Følner sequence if
n
|F |
and only if, for every ε > 0 and every finite set K, Fn is eventually (K, ε)-invariant.
NORMAL SETS IN AMENABLE SEMIGROUPS
7
We will say that a sequence of proper tilings Tk is Følner if for every finite set
K ⊂ G and every ε > 0, for large enough k, all shapes of Tk (and thus also all
tiles) are (K, ε)-invariant (in other words, by enumerating all shapes used in the
sequence of tilings we obtain a Følner sequence).
A proper tiling is called syndetic if for every shape S the center set CS is syndetic,
i.e., such that KCS = G for some finite set K (depending on S).
It is proved in [DHZ] that every countable amenable group G admits a congruent,
deterministic, Følner system of proper tilings (Tk )k≥1 . One can also easily obtain a
system of syndetic tilings with all the above properties; it suffices to let the group
act on tilings by the shifts and use any element of a minimal subsystem of the orbit
closure of any initial system of tilings. We always define T0 to be the tiling with
singleton tiles.
We are now in a position to deliver the promised “ergodic theorem for Bernoulli
measures and cylinder sets”, which seems to have been overlooked in the literature,
and which, in the context of normality, gives much better generality (as far as
Følner sequences are concerned) than the conventional pointwise ergodic theorem:
Theorem 2.3. Let (Fn )n≥1 be a Følner sequence in a countable amenable and
P∞
embeddable semigroup G, such that n=1 e−|Fn | < ∞. Let K be a finite subset of
G and let w ∈ {0, 1}K be a block over K. Then for λ-almost every x ∈ {0, 1}G one
has
|{g ∈ Fn : σg (x) ∈ [w]}|
= λ([w]).
lim
n→∞
|Fn |
Equivalently, λ-almost every x is (Fn )-generic for λ under the shift action of G
(i.e., it is (Fn )-normal).
P∞
Remark 2.4. The requirement n=1 e−|Fn | < ∞ is much weaker than that (Fn )
is tempered, and is fulfilled by the majority of reasonable Følner sequences. For
instance, it suffices that the cardinalities of the sets Fn strictly increase.
Remark 2.5. Notice that if the semigroup G is not commutative then generally
speaking, the action σg need not preserve (Fn )-normality. This annoying problem
would disappear, had we been working with the (smaller class of) two-sided Følner
sequences. The problem is however not too serous since a simple countable intersection argument implies that almost all symbolic elements are normal together with
all their images under (σg )g∈G .
Proof of Theorem 2.3. By Lemma 2.2 it suffices to consider the case where G is
a group. Given ε > 0, we will partition the group G into finitely many sets Di
(i = 0, 1, 2, . . . , r) such that for every i > 0, the set Di has positive lower density
(with respect to (Fn )) and the sets Kd with d ranging over Di are pairwise disjoint,
while the upper density of the set D0 is at most ε (it can be 0). Before we do that,
let us explain why this is sufficient.
Let n0 be such that for every n ≥ n0 the proportion of D0 in Fn is less than 2ε.
Fix an n ≥ n0 and observe the finite sequence of {0, 1}-valued random variables
defined on (X, B, λ) by
Yg (x) = 1[w] (σg (x))
with g ∈ Fn . Also, for each i = 0, 1, . . . , r, define
X
1
Yg .
Ȳi =
|Fn ∩ Di |
g∈Fn ∩Di
8
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
For i > 0, by disjointness of the sets Kd with d ∈ Di , Ȳi is the average of independent random variables, each assuming the value 1 with probability 2−|K| . By an
application of the most elementary version of the Central Limit Theorem, we have
2
λ{|Ȳi − 2−|K| | > ε} ≤ α|Fn ∩Di |ε ,
where α ∈ (0, 1) is some constant, provided |Fn ∩ Di | is large enough (which is
fulfilled simultaneously for all i if n is sufficiently large). Assume that Dr has the
smallest (still positive) lower density among all sets Di , i = 1, 2, . . . , r. Very crude
estimate by summing yields:
2
λ{∃i = 1, 2, . . . , r : |Ȳi − 2−|K| | > ε} ≤ rα|Fn ∩Dr |ε .
On the complementary set we have
|{g ∈ Fn : σg (x) ∈ [w]}|
−|K| −
2
≤ 3ε,
|Fn |
where the additional 2ε comes from taking into account the elements g belonging
to Fn ∩ D0 . We have derived that
|{g ∈ Fn : σg (x) ∈ [w]}|
2
−|K| λ −2
> 3ε ≤ rα|Fn ∩Dr |ε .
|Fn |
2
The summability assumption implies that the terms rα|Fn ∩Dr |ε form a summable
sequence. The Borel-Cantelli Lemma now yields that for λ-almost every x, the
terms
|{g ∈ Fn : σg (x) ∈ [w]}|
|Fn |
eventually remain within 3ε from 2−|K| . Since ε is arbitrary, we have proved the
desired almost everywhere convergence.
It remains to define the sets Di . To this end we will apply the tilings. As we have
mentioned earlier, G admits a congruent, deterministic, Følner system of proper,
syndetic tilings (Tk )k≥1 . For now, it suffices to apply just one proper syndetic
tiling T = (S, C) such that all its shapes S ∈ S are (K, δ)-invariant. Then, if δ
ε
is sufficiently small (precisely, not larger than |K|
), for each S ∈ S, the K-core of
S, i.e., the set SK = {g ∈ S : Kg ⊂ S} occupies at least the fraction 1 − ε of S
(see Lemma 2.6 in [DHZ]). The same is true for TK and T , where T = Sc is any
tile of T (then TK = SK c). We define D0 as the union of all sets T \ TK , over all
tiles T of T . It is not hard to see that D0 has upper density at most ε. The sets
Di (i = 1, . . . r), instead by the index i, will be parametrized by the pairs (S, g)
such that S ∈ S and g ∈ SK (since the tiling is proper, this is still a finite set of
indices). Namely, we let DS,g = gCS (recall that CS is the center set for S). Since
for each pair (S, g) as above, Kg ⊂ S (hence Kgc ⊂ Sc) and the sets Sc with c
ranging over CS are tiles (hence are pairwise disjoint), the sets Kgc are pairwise
disjoint as d = gc varies over DS,g . By syndeticity of the tiling, every center set
CS is syndetic, so is each of the sets DS,g , which easily implies its positive lower
density. Finally, its is clear that the sets DS,g together with D0 form a partition of
G. This ends the proof.
Remark 2.6. Some condition on the growth of |Fn | is necessary. For example, for
G = Z consider the Følner sequence consisting of pairwise disjoint intervals: n1
intervals of length 1 followed by n2 intervals of length 2, followed by n3 intervals
of length 3, etc. If a number nk is very large compared with k, then with very
NORMAL SETS IN AMENABLE SEMIGROUPS
9
high probability λ the restriction of x to at least one of the Følner sets of length
k will be filled entirely by 0’s. We can thus arrange the sequence nk so that the
complementary probabilities are summable over k. Then, for almost every x there
will be arbitrarily far Følner sets filled entirely with 0’s (instead of being filled nearly
half-half by 0’s and 1’s), contradicting (Fn )-genericity of x already on cylinders of
length 1. In this example, the set of (Fn )-generic elements has measure λ zero.
Remark 2.7. It is worth mentioning that the above method fails in proving the
pointwise ergodic theorem (for the Bernoulli measure) for functions other than indicator functions of cylinders. As we have already mentioned, in [AdJ], Ackoglu
and del Junco proved that in any ergodic (N, +)-action
√ the pointwise ergodic theorem along the Følner sequence of intervals√ [n, n + b nc] fails for some measurable
set. Note that the sequence e−|Fn | = e−b nc is summable, thus in the case of the
uniform Bernoulli measure, according to Theorem 2.3, this exceptional set must
not be a cylinder.
2.1. An effectively defined normal set. Although Lebesgue-almost every number is classically normal in any base b ∈ N,5 the set of computable numbers (i.e.,
whose b-ary expansion can be computed with the help of a Turing machine, like, for
example, the Champernowne number) has Lebsgue measure zero. It is so, because
the asymptotic Kolmogorov complexity of such expansions is zero, while, as shown
by A. A. Brudno [Br], a typical expansion has Kolmogorov complexity log b. Hence
computable normal numbers are highly exceptional among normal numbers. In the
construction of the binary Champernowne number there participate three types of
words:
(1) The words of length 1, 2, 3, etc. which are expansions of natural numbers.
We will call these words “bricks”. Notice that a brick never starts (on the
left) with the symbol 0, and there are exactly 2k−1 bricks of length k.
(2) The “packages”. For each k, the kth “package” is the concatenation (in
the lexicographical order) of all bricks of length k. The length of the kth
package is k2k−1 .
(3) Finally, the “chains”. For each k, the kth “chain” is formed by the packages,
from the first to the kth, concatenated together (by increase of k). The kth
Pk
chain stretches from 1 to i=1 i2i−1 .
Following this scheme, we will now describe an “effective” construction of a
Champernowne-like set in an arbitrary countable amenable group G, normal with
respect to any a priori fixed arbitrary Følner sequence (Fn ). Via Lemma 2.2 the construction applies also to (countable amenable) semigroups embeddable in groups.
We cannot claim our set to be computable, as this requires both the group and the
Følner sequence to be computable, which we do not want to assume, as this would
unnecessarily restrict generality of our result. Besides, we do not want to invoke
the computability theory, as this is not our goal. We use the term effective in
the meaning, that the construction is described by an algorithm (which practically
means an induction) allowing to determine, for every g ∈ G, whether it belongs to
the set or not, in a finite inductive step, assuming that the semigroup G and the
Følner sequence (Fn ) are known. Notice that we make absolutely no assumptions
about the Følner sequence, not even the growth condition of Theorem 2.3.
5Classically, a number is normal in base b if every word w over the alphabet {0, 1, . . . , b} occurs
in its b-ary expansion with density b−|w| .
10
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
We remark that in the classical construction of the binary Champernowne number one could introduce the following three modifications which would not destroy
the normality:
(1) one could include as bricks also the words starting with the symbol 0 (the
reason why they are not used is purely aesthetic) so that there would be
2k (rather than 2k−1 ) bricks of length k,
(2) the package of order k could contain every brick of order k repeated more
than once, as long as the number of repetitions is the same (or nearly the
same) for every brick; then the length of the package of order k would be
mk k2k for some sequence mk ,
(3) in the chain, one could repeat each package of order k more than once, say
Pk
nk times; then the length of the kth chain would equal i=1 ni mi i2i−1 .
Our construction will resemble the classical construction with the above three modifications. The reason for these changes should become clear in course of the construction.
Theorem 2.8. Let G be a countable amenable group and let (Fn ) be an arbitrary
Følner sequence in G. Then there exists an effectively defined (Fn )-normal element
x ∈ {0, 1}G .
Proof. Let (Tk )k≥0 be a congruent, deterministic, Følner system of proper, syndetic
tilings of G staring with the tiling T0 into singletons. For each k and each shape
S of Tk we create the list of all possible {0, 1}-blocks over S. Their number is 2|S|
(separately, for each S) and jointly they form a finite family. These are going to be
our “bricks” of order k. Syndeticity of each set CS and the Følner property of the
system of tilings easily imply that there exists an index r(k) such that each shape
S 0 of the tiling Tr(k) , when partitioned (in a determined way) into the tiles of Tk ,
contains, for each shape S of Tk , at least k2|S| tiles with the shape S. Now, we can
create our “packages” of order k as blocks over the shapes S 0 of Tr(k) . Namely, to
each shape S 0 we associate just one block, P (S 0 ), which satisfies the condition that,
for each shape S of Tk , P (S 0 ) contains roughly equal amounts (up to a relative
error k1 ) of all 2|S| bricks over S.
We need some terminology: for a finite set K, a block W over a finite set A is
(K, ε)-normal if A is (K, ε)-invariant and every block w over K occurs (shifted on
|
|W |
the right) in W the number of times ranging between 2|W
|K| − ε and 2|K| + ε.
The following is now easily verified:
(1) For any finite set K ⊂ G and any ε > 0, if k ≥ 1ε is large enough (so that
every shape S of Tk is (K, ε)-invariant, then every package of order k is
(A, 2ε)-normal. So is every concatenation of such (shifted) packages.
Additionally we let r(0) = 0 and we define the package of order 0 as the single
symbol 0 (to be used mainly outside the union of the sets Fn ).
The normal element x will be obtained by choosing a mixed tiling Θ selected
from the subsequence Tr(k) , i.e., a partition of G into tiles belonging to different
tilings from this subsequence. This will be possible due to congruency. Then we will
define x, by saying that, restricted to each tile of Θ, x equals the package associated
to the shape of that tile (if the considered tile belongs to the tiling Tr(k) then the
order of the package is k). In this manner x will combine packages of various orders.
The mixed tiling replaces the concept of chains: one can define the kth chain as
NORMAL SETS IN AMENABLE SEMIGROUPS
11
the part of x covered by the tiles of Θ belonging to the tilings Tr(0) , Tr(1) , . . . , Tr(k)
(and conversely, if we define a Champernowne set using chains, as a concatenation
of packages of different orders, then the tiles of Θ are simply the domains of these
packages).
It remains to describe how we define the mixed tiling Θ. The procedure will
depend on the a priori given Følner sequence (Fn ) (which so far has not affected
the construction).
For each k ≥ 1 let nk be such that all Følner sets Fn with n > nk are (Kk , k1 )−1
invariant, where Kk denotes the union of the sets S 0 S 0 over all shapes S 0 of Tr(k)
(then Fn is also (Ki , k1 )-invariant for all i ≤ k). In step 1 we begin by defining
Θ1 as T0 on the Tr(1) -saturation of the union F1 ∪ F2 · · · ∪ Fn1 . Inductively, we
define the mixed tiling Θk as Tr(k−1) on this part of the Tr(k) -saturation of the
union F1 ∪ F2 · · · ∪ Fnk on which the tiling has not yet been defined. Since Θk−1 is
defined on a union of complete tiles of Tr(k−1) , there is no collision now. When this
induction is finished, we additionally define Θ as T0 on the yet untiled reminder of
the group (if it exists).
Observe that the mixed tiling Θ satisfies the follwing:
−1
(2) Each set Fn is covered only by tiles of such shapes S 0 that Fn is (S 0 S 0 , k1 )invariant (where k is the largest such that n > nk ). This implies that Fn
differs from its Θ-saturation by at most k1 |Fn | elements.
(3) For each k ≥ 1, Θ uses only finitely many tiles belonging to Tr(k) .
As we have already described earlier, Θ determines x. It remains to verify the
normality of x. Let K ⊂ G be a finite set and fix some ε > 0. It suffices to show
that, for n sufficiently large, x|Fn is (K, 4ε)-normal. Let k0 ≥ 1ε be such that for
each k ≥ k0 every shape S of Tk is (K, ε)-invariant. By (1), every package of order
k ≥ k0 is (K, 2ε)-normal. Pick any n > nk0 . For estimating K-normality of x on
Fn we first replace Fn by its Θ-saturation, which, by (2), affects the estimation by
at most ε, then we remove from this saturation all (finitely many) tiles of orders
smaller than k0 . If n is large enough, this last step also affects the estimation by
at most ε. Now we are left with a set on which x is a concatenation of packages of
orders at least k0 , so it is (K, 2ε)-normal. This ends the proof.
3. Multiplicative normality
In this section we will focus on the action of (N, ×) on the symbolic space {0, 1}N .
In this case the shift action, henceforth called the multiplicative shift and denoted
by (ρn )n∈N is defined on {0, 1}N as follows:
if x = (xj )j∈N then ρn (x) = (xjn )j∈N .
In other words, ρn maps each binary sequence to its subsequence obtained by
reading its every nth term. Clearly, the classical ( 12 , 12 )-Bernoulli measure on the
symbolic space {0, 1}N is invariant under both the additive and multiplicative shift
actions, and in both cases it is the unique measure of maximal entropy. In fact we
are dealing here with the case of a sequence of independent identically distributed
random variables, which corresponds to the Bernoulli process regardless of the
applied action, as long as the action “permutes” the indices (we use quotation
marks, because our “permutations” are not surjective). Note that both shift actions
are ergodic (in fact mixing) and their Kolmogorov-Sinai entropies equal the entropy
12
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
of the generating partition {[0], [1]}, i.e., to log 2, and so are the topological entropies
of both shift actions.
3.1. Følner sequences in (N, ×). The semigroup (N, ×) is a free Abelian semigroup generated by the set of primes. We will denote the set of primes by P and
view (N, ×) as the direct sum6
M
G=
Np ,
p∈P
where, for each p ∈ P, Np is the same additive semigroup (N ∪ {0}, +). The
isomorphism is given by
(k1 , k2 , . . . , kr ) 7→ pk11 pk22 · · · pkr r ,
where P = {p1 , p2 , . . . } = {2, 3, 5, . . . } is the natural ordering of the primes. Notice
that G is additive, i.e., in this representation multiplication of natural numbers is
interpreted as addition of vectors.
In order to deal with the actions of (N, ×) it is crucial to identify convenient
choices of Følner sequences in this semigroup. A natural choice of a Følner sequence
(Fn )n∈N in G is given by rectangular boxes containing the origin:
(n)
(n)
(n)
Fn = {0, 1, . . . , k1 } × {0, 1, . . . , k2 } × · · · × {0, 1, . . . , kdn } × {0} × {0} × · · · .
For convenience, we will skip the singleton components. The parameter dn will
(n)
be referred to as the dimension of Fn . The number ki will be called the size of
Fn in the ith direction. The dimensions of the Følner sets tend to infinity, and so
(n)
do the sizes in each given direction (i.e., ki go to infinity for each i). Any such
Følner sequence will be called anchored rectangular. The verification of the Følner
property is straightforward. Preferably, the Følner sets should increase with respect
(n)
to inclusion, which means that the sequences (dn ) and (ki ) for each i should be
(n)
(n)
(n)
nondecreasing and the sum k1 + k2 + · · · + kdn should be increasing. Such
increasing Følner sequences will be called nice. Not every nice Følner sequence
(Fn ) is tempered, however, since the cardinalities |Fn | strictly increase, Theorem
2.3 applies.
Every nice Følner sequence occurs as a subsequence of a specific “nice and slow”
(n)
(n)
(n)
Følner sequence, such that in each step the sum k1 + k2 + · · · + kdn increases
only by 1. The choice of a nice and slow Følner sequence is equivalent to fixing a
“sequence of directions” (in ), in which every natural value appears infinitely many
(n)
times, and letting ki equal the number of times i occurs in the finite sequence
i1 , i2 , . . . , in . There are several fairly natural possible choices of the sequence (in ),
for example:
• 1; 1, 2; 1, 2, 3; 1, 2, 3, 4; 1, 2, 3, 4, 5; . . . the staircase type,
• 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, . . . the Toeplitz type.
Now we can translate the above to the multiplicative representation (N, ×). In
(N, ×) we have the natural partial order given by M < m ⇐⇒ m|M . The set N
equipped with this order is a directed set (i.e., every two elements have a common
upper bound; in this case a common multiple), thus it makes sense to say that a
sequence (Ln ) multiplicatively tends to infinity, or that it multiplicatively increases
6Recall that a direct sum is the subset of the Cartesian product (with addition acting coordinatewise) consisting of points with at most finitely many nonzero coordinates.
NORMAL SETS IN AMENABLE SEMIGROUPS
13
to infinity. Now, any anchored rectangular (resp. nice) Følner sequence is obtained
by fixing some sequence (Ln ) which multiplicatively tends (resp. multiplicatively
increases) to infinity, and letting Fn be the set of all numbers multiplicatively
smaller than or equal to Ln (i.e., Fn is the set of all divisors of Ln ). We will call
Ln the leading element of Fn and we will refer to Fn as the rectangle ending at Ln .
is a prime, for every n. Notice
A nice and slow Følner sequence is obtained if LLn+1
n
that even if (Fn ) is nice and slow, the cardinalities |Fn | grow relatively fast. Indeed,
from time to time the dimension dn+1 of Fn+1 has to increase, i.e., a new direction
has to be included, and then the cardinality doubles: |Fn+1 | = 2|Fn |. Otherwise
the cardinality is multiplied by a factor smaller than 2, but in any case a whole
rectangle (of dimension dn − 1) is added. In particular, |Fn+1 | − |Fn | > 1 for n > 1.
Obviously, there are many other Følner sequences in G: the rectangular boxes
need not be anchored at zero, and moreover, they can be replaced by other shapes.
For instance, it is possible to create a Følner sequence with |Fn | = n, but it is not
going to be rectangular (however, it may have a nice and slow Følner subsequence).
Since this is just a remark, we skip further details. Let us agree that anchored
rectangular and, in particular, nice Følner sequences form the most natural family
(mainly due to the good multiplicative interpretation). Unfortunately, there seems
to be no preferred “canonical” choice.
3.2. Multiplicative Champernowne set. The construction of a Champernowne
set in (N, ×) can be made significantly more transparent than in the general case.
This is due to the fact that this semigroup admits a system of (congruent, deterministic, Følner, syndetic) monotilings, i.e., tilings with only one shape. In fact,
any rectangular box tiles the semigroup, while a congruent system of tilings is obtained from a specific Følner sequence, which we will call doubling. This will enable
us to create “condensed” packages which contain every brick exactly once (like in
the classical Champernowne construction). In a chain we still must repeat each
package more than once (this we would have to do even in the two-dimensional
semigroup (N2 , +)), but we will use the least possible number repetitions, to fill a
rectangular box the size of the next order package. In this manner we will obtain
a “compendious” Champernowne set, which will turn out to be normal at least
with respect to the same doubling Følner sequence which we used in its construction. Later we will present a slightly less compendious modification of the same
construction, which produces a “net-normal” set, i.e., normal with respect to any
nice Følner sequence.
We begin by formalizing the new type of Følner sets. Again, we will interpret
(N, ×) as the additive semigroup G.
Definition 3.1. A nice Følner sequence (Fn ) is called doubling if Fn+1 is a disjoint
union Fn ∪ (vn + Fn ) (for some vn ∈ G). Note that since Fn+1 is still rectangular,
vn must be equal to one of the axial vectors spanning Fn .
Any doubling Følner sequence can be obtained by the following procedure. As
before, in the construction of a nice and slow Følner sequence, we fix a sequence
of directions (in )n≥1 in which each natural i is repeated infinitely many times. We
begin with the “zero Følner set” F0 = {0}. Once the Følner set Fn is determined,
the next one, Fn+1 , instead of growing in by a unit in the direction in+1 , is doubled
in that direction. The volume of Fn will hence be equal to 2n . For example, if (in )
14
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
is the staircase sequence 1; 1, 2; 1, 2, 3; . . . , the first Følner sets are
F0 = {0}
F1 = {0, 1}
F2 = {0, 1, 2, 3}
F3 = {0, 1, 2, 3} × {0, 1}
F4 = {0, 1, 2, 3, 4, 5, 6, 7} × {0, 1}
F5 = {0, 1, 2, 3, 4, 5, 6, 7} × {0, 1, 2, 3}
F6 = {0, 1, 2, 3, 4, 5, 6, 7} × {0, 1, 2, 3} × {0, 1}
..
.
The construction. We shall now construct an (Fn )-normal element c ∈ {0, 1}N
for a doubling Følner sequence. We describe how we build the generations of bricks
and packages, and how we place these in c (here we will use the language of chains
rather than mixed tilings). The bricks of order k will simply have the shape of
the Følner set Fk (and volume 2k ). We accept as bricks of order k all blocks over
k
Fk . Thus there will be 22 different bricks, which, concatenated together (each
k
used exactly once), produce a package of the volume 22 +k . Because the sizes in
all directions of all our objects (Følner sets, bricks, packages, etc.) are powers of
2, each smaller one divides each larger one. Thus we can arrange the package so
that its shape matches F2k +k (the index 2k + k plays the role of r(k) from the
general construction). We assume inductively that the (k −1)st chain has exactly
the same shape as the package of order k (we skip this assumption in step 0 while
in step 1 it holds; see also Figure 1 below). This assumption replaces (and realizes)
the requirement from the general construction that the (k−1)st chain is saturated
with respect to the tiling number rk , so it “fits” to be concatenated together with
packages of order k. Now we can build the kth chain. It consists of: the (k −1)st
chain occupying the “lower left” corner, i.e., containing the origin (this does not
k
apply to step 0), and 22 +1 − 1 (in step 0 we do not subtract 1) shifted copies of
k+1
the kth order package, so that the chain has volume 22 +k+1 (in step zero this
volume equals 8). The chain can be arranged to have the shape of F2k+1 +k+1 , i.e.,
the same as that of the package of order (k + 1), as required in the induction.
Figure 1 below is made for the (mentioned above) “staircase type” doubling
Følner sequence and it shows: the package of order 0 with the initial “zero” brick
of order 0 shaded, and next to it the 0th chain, which is a concatenation of four
packages. Below that we show the package of order 1 with the initial “zero” brick
of order 1 shaded, and next to it the 1st chain which is a concatenation of the
preceding chain (shaded) and seven identical packages of order 1. The last picture
shows the package of order 2 with the initial “zero” brick of order 2 shaded. The 2nd
chain is too large to be shown. It is a concatenation of the 1st chain and 31 copies
of the package of order 2, and it fits the shape of F11 (which is four-dimensional).
Eventually the chains converge to a complete (infinite-dimensional) element c ∈
{0, 1}G . The subset of N of which c is the indicator function will be called the
multiplicative Champernowne set. Its disadvantage is that it is normal with respect
to a specific doubling Følner sequence (Fn ). This will be fixed in our next, more
universal (though less compendious) construction.
NORMAL SETS IN AMENABLE SEMIGROUPS
0 1
15
0 1 0 1
0 1 0 1
1 0 1 1
0 0 0 1
1 0 1 1 1 0 1
0
1 0 10 10 11 00 10 10
0 01 00 11 01 01 00 11
0 10 00 10 11 00 10 10
0 1 0 1 0 0 0 1
1 1 1 0 1 1 1
0 11 11 00 01 11 11 11
0 11 00 11 00 11 10 01
0 01 10 00 00 01 10 10
0 0 0 0 0 0 0 1
1
1
1
1
1
0
1
1
Figure 1
We need to check (Fn )-normality of c. Recall that the general construction used
in fact two Følner sequences in the group G: first one, denoted (Fn ), which was
arbitrary and given a priori, and second one (implicit, not denoted)—the shapes of
the system of tilings (recall that the system of tilings was assumed to be Følner),
independent of the former. The construction of the packages depended only on the
tilings, while (Fn ) affected the construction of the chains. Our current construction
fits the general scheme, where the doubling sequence (denoted (Fk )) is responsible
for the system of tilings. What we want now is to see whether the same Følner
sequence (now denoted (Fn )) is also good in the other role. Recall that given a
finite set K and ε > 0, high order packages are multiplicatively (K, ε)-normal. So,
to prove (Fn )-normality of c we need to check that if n is large, majority of Fn is
concatenated of complete packages of some high orders. This is certainly true for
n = 2k + k, as then Fn is filled with the kth chain, consisting of one (k−1)st chain
and many complete packages of order k (of the same size as the (k −1)st chain).
Then, Fn+1 contains a concatenation of the (k − 1)st chain (which is good) and
one package of order k + 1, so it is also good. Similar argument applies to Fn+2 ,
which is filled by the (k−1)st chain and three packages of order k. We continue this
reasoning for Fn+3 , Fn+4 , . . . until F2k+1 +k+1 , when we can use the first argument
for k increased by 1.
Remark 3.2. We can also deduce multiplicative normality of c with respect to the
nice and slow Følner sequence of which (Fk ) is a subsequence (to obtain such a nice
and slow Følner sequence, instead of doubling a direction we increase it by 1 several
times). We omit the details. On the other hand, c is definitely not nor normal for
some other nice Følner sequences. For instance, if the Følner sets increase in the
first direction much faster than in other directions (elongated shapes) then the
symbol 0 will prevail. We skip the details again.
3.3. Net-normal sets. The multiplicative order < in (N, ×) (see Section 3.1) is
not linear, however, (N, <) is a directed set (a net). Thus, although the family of
all anchored rectangular boxes does not form a Følner sequence, it can be viewed
16
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
as a net (ordered by inclusion) if one labels each box by its leading element. For
short, we will call it the Følner net (skipping “of anchored rectangular boxes”).
Definition 3.3. A set A in (N, ×) (or a {0, 1}-sequence xA ) is net-normal if
it is multiplicatively normal with respect to each nice Følner sequence in (N, ×).
Equivalently, A is net-normal if for every block w over a finite set K, the density,
along the Følner net, of the set V (w, A) of multiplicative occurrences of w in xA ,
exists and equals 2−|K| .
Remark 3.4. Yet another equivalent definition of net-normality is obtained by demanding normality with respect to each anchored rectangular Følner sequence.
An interesting question (which has no counterpart in the additive context) concerns the existence of net-normal sets and their typicality. We will show in Theorem 3.5 that net-normal sets exist, and, in Theorem 3.8, that they are exceptional.
(In Section 4.5 we will establish some interesting combinatorial properties of such
sets.)
Theorem 3.5. There exists an effectively defined net-normal set A ⊂ (N, ×).
Proof. As in the construction of the multiplicative Champernowne set, we view
(N, ×) as the additive semigroup G. The construction will be just a slight modification of that presented in Section 3.2. The bricks and packages will be the same
(they depend on the choice of the doubling sequence (Fk )). Only the mixed tiling
will be different—the resulting chains will become infinite. We will be using the
following observation:
Lemma 3.6. Fix some finite set K ⊂ N and ε > 0. Let Z(K, ε) denote the union of
all anchored rectangular boxes which are not multiplicatively (K, ε)-invariant. Then
Z(K, ε) has density zero with respect to any (not necessarily rectangular) Følner
sequence (Fn ) in (N, ×).
Proof. For sake of convenience we will write [0, n] instead of {0, 1, 2, . . . , n}. Intuitively, a rectangular box is not multiplicatively (K, ε)-invariant if it is “narrow” in
some direction represented in K, and narrow sets have density zero. More precisely,
let K̄ = [0, k1 ] × [0, k2 ] × · · · × [0, kq ] be the smallest rectangular box containing
K. If a rectangular box B = [0, b1 ] × [0, b2 ] × · · · × [0, br ] is not (K, ε)-invariant,
all the more it is not (K̄, ε)-invariant, i.e., there is an index i ∈ {1, 2, . . . , q} such
. Thus, the union Z(K, ε) of all such rectangles B is
that bi < αi , where αi = 2kε i S
q
contained in the finite union i=1 Xi , where Xi is the set of all vectors in G whose
ith coordinate is smaller than αi . It is clear that each set Xi has density zero
with respect to any Følner sequence (Fn ): for any δ > 0 the sets Fn are eventually
(αi ei , δ)-invariant where ei is the ith versor, while Xi is disjoint from αi ei + Fn . With this lemma in hand we can describe the modification of the mixed tiling
Θ (or, equivalently, of the chains). Namely, in step k, we cover with the packages
of order k the yet uncovered part of the saturation, with respect to the tiling by
F2k+1 +k+1 (i.e., Trk+1 in the general construction), of Z(Fk , k1 ). Congruency implies
that the tiling Θ is well defined and it determines an element x = xA . It remains
to verify multiplicative normality of x with respect to any nice Følner sequence.
The argument is identical as that for the element c in the preceding section, except
one detail: this time Θ uses infinitely many tiles from Tr(k) . But these tiles cover
only the Tr(k+1) -saturation of a set which, by Lemma 3.6, has universal (i.e., with
NORMAL SETS IN AMENABLE SEMIGROUPS
17
respect to any Følner sequence) density zero (and clearly so does its saturation with
respect to any proper tiling). The rest of the proof of normality of x is identical as
before.
Remark 3.7. Notice that, for any countable amenable (semi)group G, it is impossible to find a symbolic element x ∈ {0, 1}G which is normal with respect to
all Følner sequences. For instance, if A is (Fn )-normal for some Følner sequence
(Fn ) in (N, +), then its complement Ac contains arbitrarily long intervals, which
constitute a Følner sequence disjoint from A. (A similar argument applies to any
amenable semigroup.)
Theorem 3.8. The collection of all net-normal symbolic elements in {0, 1}G has
measure zero for the Bernoulli measure λ.
Proof. Let x be a typical point for this measure. We start by observing the “zero
rectangle” {0} (which more precisely should be written as {0} × {0} × {0} × · · · ).
Any value x0 ∈ {0, 1} is admitted in x at this single position. Next, we can “double”
this rectangle in countably many directions, that is, we can create rectangles
{0} × {0} × {0} × · · · × {0} × [0, 1] × {0} × · · · ,
where the interval [0, 1] appears at any possible position k in the product. With
probability 1, for some k = k1 (x), the symbol seen on the added coordinate will
be 0. Next, we double this rectangle in the direction dictated by the “staircase
sequence of directions”, that is we create the rectangle
[0, 1] × {0} × {0} × · · · × {0} × [0, 1] × {0} × · · ·
(the first term of the “staircase sequence” instructs us to double the first coordinate). This time some two “random” symbols are added. But in the next step
we double the rectangle again in all possible directions larger than k1 (x), i.e., we
create rectangles
[0, 1] × {0} × {0} × · · · × {0} × [0, 1] × {0} × · · · × {0} × [0, 1] × {0} × · · · ,
where the third interval [0, 1] occurs at some k ≥ k1 (x), and again, with probability 1, for some k = k2 (x), the symbols observed at the two added positions will
be zeros. Alternately, in even steps we double the rectangles as dictated by the
staircase sequence, in odd steps we double them so that all new symbols (which
constitute half of the volume) are zeros. It is clear that eventually, for almost every
x, we will obtain a doubling Følner sequence (Fn ) (depending on x), such that x is
1
not (Fn )-normal (the upper (Fn )-density of zeros is at least 21 + 81 + 32
+ · · · ). 4. Combinatorial and Diophantine properties of additively and
multiplicatively normal sets in N
In this section we will be focusing on the combinatorial and Diophantine richness
of normal sets in (N, +) and (N, ×). Before starting the discussion we review some
terminology (and introduce some new terms, too).
(i) We say that a set S ⊂ N is additively (multiplicatively) large if for some Følner
sequence (Fn ) in (N, +) (resp. (N, ×)) we have d(Fn ) (S) > 0.
(ii) A set S in a semigroup G is called thick if it contains a right translate of
every finite set. The family of thick sets in G is denoted by T (G). Note
that S ∈ T (N, +) if and only if S contains arbitrarily long intervals, and
18
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
that S ∈ T (N, ×) if and only if S contains arbitrarily large sets of the form
an {1, 2, . . . , n} = {an , 2an , . . . , nan }.
(iii) We say that S ⊂ N is additively normal if it is (Fn )-normal for some Følner
sequence (Fn ) in (N, +), while if (Fn ) = ({1, 2, . . . , n}), we will call S a classical normal set. Similarly, a set S is called multiplicatively normal if it is
(Fn )-normal for some Følner sequence (Fn ) in (N, ×)
(iv) Sets of the form F S(ni )∞
i=1 = {ni1 + ni2 + · · · + nik ; i1 < i2 < · · · < ik , k ∈ N}
(F P (ni )∞
=
{n
n
·
·
· nik ; i1 < i2 < · · · < ik , k ∈ N}) are called additive
i
i
1
2
i=1
(resp, multiplicative) IP-sets.
4.1. Multiplicative versus additive density and normality – some basic
observations. Recall that (Fn )-normality (addive or multiplicative) of a symbolic
element x is defined as the property that every block w over a finite set of coordinates K occurs in x with (Fn )-density 2−|K| . Note, however, that the term “occurs” has different meaning in the additive and multiplicative setups. For example,
if w is a word over {1, 2, . . . , k}, it additively occurs at a position n of some x if
x|{n+1,n+2,...,n+k} = w, while it multiplicatively occurs at n if x|{n,2n,3n,...,kn} = w.
Theorem 2.3 implies that for any Følner sequence (Fn ) in (N, +) such that |Fn |
increases, λ-almost every x is (Fn )-normal and, similarly, for any Følner sequence
(Kn ) in (N, ×) such that |Kn | increases, λ-almost every x is (Kn )-normal. So λalmost every x is both additively (Fn )-normal and multiplicatively (Kn )-normal.
On the other hand, the two notions of normality are in some sense orthogonal: additively normal sequences can be multiplicatively trivial (have multiplicative density
0 or 1), and vice-versa, multiplicatively normal numbers can have additive density
0 or 1. More precisely, the following holds.
Theorem 4.1. For any Følner sequence (Kn )n∈N in (N, ×) there exists a set A ⊂ N
with (Kn )-density 1 and having universal additive density 0 (here universal means
that the additive density can be computed with respect to an arbitrary Følner sequence in (N, +)).
Proof. First observe that for any m the set mN (being a multiplicative shift of a
set of (Kn )-density 1) has (Kn )-density 1. Let nm be such that for each n > nm
1
. The set A is defined
the fraction of multiples of m! in Kn is larger than 1 − m
as the union K1 ∪ K2 ∪ · · · ∪ Kn2 to which we add all multiples of 2! contained in
the union Kn2 +1 ∪ Kn2 +2 ∪ · · · ∪ Kn3 , all multiples of 3! contained in the union
Kn3 +1 ∪ Kn3 +2 ∪ · · · ∪ Kn4 , etc. It is obvious, that the (Kn )-density of A equals 1.
On the other hand, since A has increasing gaps, its additive density is equal to 0
(for any additive Følner sequence).
Note that the complementary set Ac has (Kn )-density 0 and universal additive
density 1. Further, given a Følner sequence (Fn ) in (N, +), let B ⊂ N be a set which
is both multiplicatively (Kn )-normal and additively (Fn )-normal. Then A ∩ B is
multiplicatively (Kn )-normal, while it has universal additive density zero, and on
the other hand, Ac ∩ B is additively (Fn )-normal and has multiplicative (Kn )density 0. These examples justify our claim above that the notions of multiplicative
and additive normality are in “general position”.
Remark 4.2. A statement symmetric to Theorem 4.1, in which one fixes a Følner
sequence (Fn ) in (N, +) (for instance the classical one) and seeks a set of (Fn )density 1 and universal multiplicative density 0, does not hold. As a matter of
NORMAL SETS IN AMENABLE SEMIGROUPS
19
fact, any set of upper density 1 with respect to the classical Følner sequence in
(N, +) has density 1 with respect to some Følner sequence in (N, ×). Indeed, in
the proof of [BM, Theorem 6.3] it is shown that if A has classical upper density 1,
so does A/n ∩ A for every n (see Definition 4.14 below). By an obvious iteration,
we get that A ∩ A/2 ∩ A/3 ∩ · · · ∩ A/n is nonempty, which implies that A contains
arbitrarily large sets of the form an {1, 2, . . . , n}, i.e., A is multiplicatively thick (see
(ii) above). This, in turn, implies that for some Følner sequence (Kn ) in (N, ×) one
has d(Kn ) (A) = 1.
On the other hand, there are sets A ⊂ (N, +) with d(A) = 1 − ε such that A has
universal multiplicative density zero (take for example all numbers not divisible by
some large n).
4.2. Elementary combinatorial properties of additively and multiplicatively normal sets. The above Theorem 4.1 (and also the “lack of symmetry”
observed in Remark 4.2) allows us to expect that, in general, the combinatorial
properties of additively and multiplicatively normal sets are distinct (we will see
below that this is indeed the case). Some properties of additively/multiplicatively
normal sets follow from the fact that they are additively/multiplicatively large and,
moreover, additively/multiplicatively thick.7
Hence, for example, any additively/multiplicatively normal set contains an additive/multiplicative IP-set (respectively).8 Now, IP-sets can be defined as solutions
of (an infinite) system of linear equations, and in our quest for patterns in normal
sets, it is natural to inquire which Diophantine equations and systems thereof are
always solvable in normal sets. The following two theorems shed some light on this
question:
Theorem 4.3. If S is a multiplicatively normal set then any homogeneous system
of finitely many polynomial equations (with several variables) which is solvable in
N is solvable in S.
Proof. Since S ∈ T (N, ×) (i.e., is multiplicatively thick), S contains arbitrarily long
sets of the form an {1, 2, . . . , n}. If a given homogeneous system is solvable in N
then it is solvable in {1, 2, . . . , n} for some n and hence, due to homogenincity, also
in an {1, 2, . . . , n}.
Remark 4.4. Note that it follows from Theorem 4.3 that any multiplicatively normal
set contains arbitrarily large finite sums sets F S(ni )m
i=1 = {ni1 +ni2 +· · ·+nik ; i1 <
i2 < · · · < ik , k ∈ {1, 2, . . . , m}} (indeed, these sets can be described as solutions of
finite homogeneous systems of linear equations). We will show now that in general
multiplicatively normal sets need not contain infinite finite sums sets (which are
also called IP-sets) or shifts thereof.
Take any Følner sequence (Fn ) in (N, ×) and let A be the set of (Fn )-density
1 constructed in the proof of Theorem 4.1 (A has universal additive density zero).
For each n, this set contains only finitely many numbers not divisible by n. On
the other hand, it is well known (and also easy to see) that every additive IP set
contains, for arbitrarily large n, infinitely many numbers divisible by n as well as
7Actually, it is nearly obvious that a combinatorial configuration occurs in every additively/multiplicatively thick set if and only if it appears in every additively/multiplicatively normal
set.
8It is well known and easy to prove that any thick set contains an IP-set.
20
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
infinitely many numbers not divisible by n. Thus any (shifted or not) IP-set contains
infinitely many numbers not divisible by n, and hence cannot be contained in A.
→
Theorem 4.5. Let A x = 0 be a partition regular system of finitely many linear
equations with n variables. Then, for any additively normal set S one can find a
→
solution x = (x1 , x2 , . . . , xn ) with all entries in S.
Proof. The proof is short but uses same facts from Ramsey theory, topological
dynamics and topological algebra in the Stone–Čech compactification βN viewed
as a semitopological semigroup obtained by an extension of the operation in (N, +).
Since this theorem forms only a rather small fragment of a big picture, in order
to save space, we will be using some terms and results without providing all the
needed details (but remedying this by providing pertinent references).
First, note that our additively normal set is thick and hence is a member of a
minimal idempotent in (βN, +). Further, any member of a minimal idempotent in
(βN, +) is a central set (see, for example, Definition 5.8 and Lemma 5.10 in [Be5]).
Now it only remains to invoke the theorem due to Furstenberg which states that
→
any central set contains solutions to any partition-regular system A x = 0 ([Fu2,
Theorem 8.22]).9
One can actually show that a system of linear equations is partition regular if
and only if it is solvable in any additively normal set (equivalently, in any thick
set). For sake of simplicity we prove this equivalence in the case of one equation
with three variables. Note that any such equation (which has at least one solution)
can be written as ia + jb = kc with i, j, k ∈ N.
Theorem 4.6. Let i, j, k be three natural coefficients. The following conditions are
equivalent:
(1) k ∈
/ {i, j, i + j},
(2) the equation ia + jb = kc is partition regular,
(3) the equation ia + jb = kc is solvable in any thick set,
(4) the equation ia + jb = kc is solvable in any additively normal set.
Proof. Equivalence of (1) and (2) is well known. As a matter of fact, a necessary and
sufficient condition for partition regularity of equation i1 a1 +i2 a2 +· · ·+in an = 0 is
that some subset of coefficients sums up to zero. See, for example [GRS]. Conditions
(3) and (4) are equivalent
S since every additively normal set is thick, while every thick
set contains the union n Fn of some Følner sequence and—within this union—an
additively normal set. Solvability of partition regular linear equations in thick (and
hence additively normal) sets is our Theorem 4.5. It remains to consider coefficients
for which (1) does not hold and construct a thick set A ⊂ N, which contains no
solutions, i.e., is such that (iA + jA) ∩ kA = ∅.
Since k ∈
/ {i, j, i + j} there exist a rational number δ > 0 such that
k[1, 1 + δ] ∩ (i[1, 1 + 2δ] ∪ j[1, 1 + 2δ] ∪ (i + j)[1, 1 + δ]) = ∅.
Let
A=
∞
[
In , where In = rn [1, 1 + δ],
n=1
9See [Fi2, Theorem 4.1] for a more general result of this kind, obtained by a different method.
NORMAL SETS IN AMENABLE SEMIGROUPS
21
where the numbers rn ∈ N are such that rn δ ∈ N and grow geometrically with a
large ratio. Obviously, A is a thick set. Choose any a, b ∈ A. If a, b belong to the
same interval In , then ia + jb ∈ (i + j)rn [1, 1 + δ]. If a, b belong to two different
intervals, say Im , In with m < n, then, since rm is much smaller than rn , ia + jb
is either in irn [1, 1 + 2δ] or in jrn [1, 1 + 2δ]. In any case, ia + jb ∈
/ kIn and, due
to the fast growth of rn , ia + jb ∈
/ kIl for any other l. So, (iA + jA) ∩ kA = ∅, as
needed.
Remark 4.7. We will show later (see Corollary 4.18) that any classical normal set A
has the stronger property that any equation ia + jb = kc with i, j, k ∈ N is solvable
in A.
We conclude this section with a simple observation that additively normal sets
always contain at least some modest amount of multiplicative structure.
Theorem 4.8. Any additively normal set A contains arbitrarily large “product
sets” of the form {y1 , y1 y2 , . . . , y1 y2 . . . yk }.
Proof. The result follows from the (almost obvious) fact that any thick sets contains
arbitrarily large product sets.
4.3. Covering property of translates of normal sets. The special case (for
(Z, +)) of the following result is implicit in [BW]. We give a short proof for arbitrary
countable amenable groups (and embedable semigroups).
Lemma 4.9. Let G be a countable amenable group in which we fix arbitrarily a
Følner sequence (Fn ). If A is an (Fn )-normal set and B ⊂ G is infinite, then the
set BA has (Fn )-density 1.
Proof. Observe that if K ⊂ G is finite then KA has density precisely 1 − 2−|K| .
Indeed, g ∈
/ KA is equivalent to K −1 g∩A = ∅, i.e., the indicator function xA |K −1 g =
0. By normality of A, the last equality holds for elements g whose density is 2−|K| .
If B is infinite, the lower density of BA is larger than 1 − 2−k for any k, so BA has
density 1.
Corollary 4.10. Lemma 4.9 holds in countable amenable semigroups embeddable
in groups, in particular in (N, +) and (N, ×). Moreover, we also have that B −1A
defined as the set of such g ∈ G that bg ∈ A for some b ∈ B, has (Fn )-density 1.
Proof. See Lemma 2.2.
Remark 4.11. The following useful observation generalizes [BW, Theorem 2]: If
C ⊂ G has positive upper (Fn )-density then bA ∩ C has (Fn )-density zero for at
most finitely many b ∈ N. Indeed, if k is such that d(Fn ) (C) > 2−k , then for any
K ⊂ N with |K| = k one has d(Fn ) (KA ∩ C) > 0. If there were k different elements
b1 , . . . , bk ∈ N satisfying d(Fn ) (bi A ∩ C) = 0, then the set K = {b1 , . . . , kk } would
violate the last inequality.
Example 4.12. The following example, in the classical setup of (N, +), shows that for
an additively (in particular, classical) normal set A the complement of B + A need
not be finite, even if B = A. Start the construction by choosing a finite word w1 with
good normality properties. Let w̃ denote the block obtained from w by switching
zeros and ones and writing the symbols in reverse order. The concatenated word
v1 = w1 w̃1 also has good normality properties and is antisymmetric, i.e., it satisfies
22
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
v1 (k) = 1 − v1 (n1 − k), for all 0 ≤ k < n1 , where n1 is the length of v1 . Note that if
A is any set whose indicator function xA starts with v1 then A+A misses n1 . Let w2
be a word much longer than w1 and with much better normality properties. Define
v2 as the concatenation v1 w2 w̃2 v1 . This word starts with v1 , is antisymmetric and
has nearly as good normality properties as w2 . If xA starts with v2 then A + A
misses both n1 and n2 = |v2 |. Continuing in this fashion we will end up with an
infinite set A which is normal and such that A + A misses infinitely many integers.
Remark 4.13. On the other hand, thickness alone easily implies that A − A = N
(where A − A is understood as understood as the set of positive differences of
elements from A).
4.4. Divisibility of classical normal sets. First applications. From now on,
we will be dealing with classical normal sets in (N, +) (and also, briefly, with netnormal sets in (N, ×)). As we will see they exhibit especially rich combinatorial
structure (not shared by general additively or multiplicatively normal sets). In this
subsection we focus on general linear equations with three variables in classical
normal sets.
Definition 4.14. Given A ⊂ N, and n ∈ N, denote by A/n the set {m : nm ∈ A}
(formally, this is n1 (A ∩ nN), or invoking the multiplicative shift, xA/n = ρn (xA )).
Lemma 4.15. If A is a classical normal set, so is A/n for any n ∈ N.
Remark 4.16. (i) Lemma 4.15 implies that (xk ) ∈ {0, 1}N is classical normal if
and only if so is the sequence (xkn ), for every n ∈ N. This result was proved
(by a different method) in D. Wall’s thesis [Wa]. For convenience of the reader,
we provide an ergodic proof.
(ii) A nontrivial fact which is implicitly used in the proof is the divisibility property
of the classical Følner sequence Fn = {1, 2, . . . , n} in (N, +): for any k ∈ N,
(Fn /k) is essentially the same Følner sequence. For example, for k = 3, (Fn /k)
starts with ∅, ∅, F1 , F1 , F1 , F2 , F2 , F2 , F3 , F3 , F3 , . . . .
Proof of Lemma 4.15. Suppose A/n is not normal, i.e., some word w having length
k does not occur in the indicator function xA/n of A/n with the correct frequency
2−k . This means that the “scattered” block ŵ (in which the entries of w appear
along the arithmetic progression {n, 2n, . . . , kn}) occurs in xA starting at coordinates m ∈ nN with, say, upper density different from 2k n1 . Let y be the periodic
sequence y(i) = 1 ⇐⇒ n|i. Now consider the pair (xA , y) (it is convenient to
imagine this pair as a two-row sequence with xA written above y). This pair is a
sequence over four symbols in {0, 1}2 . The above means that the scattered block
(ŵ, v̂) where v̂ denotes the block of just 1’s at each bottom position {n, 2n, . . . , kn},
occurs in (xA , y) with the upper density different from 2k n1 . There is a subsequence
ni such that the upper density is achieved along intervals {1, . . . , ni }, and moreover,
the corresponding sequence of normalized counting measures supported by the sets
{(xA , y), σ((xA , y)), σ 2 ((xA , y)), . . . , σ ni ((xA , y))}
(where σ((x, y)) = (σ(x), σ(y))) converges in the weak-star topology to a shiftinvariant measure µ on ({0, 1}2 )N . Since xA is normal (i.e., generic for the Bernoulli
measure λ) and y is periodic (hence generic for the unique invariant probability
measure ξ on the periodic orbit of y), the marginal measures of µ are λ and ξ. Now,
µ([ŵ] × [v̂]) 6= 2k n1 = µ([ŵ])ξ([v̂]), which means that µ 6= λ × ξ. This contradicts
NORMAL SETS IN AMENABLE SEMIGROUPS
23
disjointness of Bernoulli measures from periodic measures (which is a particular
case of disjointness between K-systems and entropy zero systems, see [Fu1]).
We can now derive another fact in the classical case:
Theorem 4.17. If A is a classical normal set and n, m ∈ N are coprime, then
nA ± mA (restricted to N) has density 1.
Proof. The theorem follows from the fact that, relatively, in every residue class
mod n, (i.e., in the set nN + i for each i = 0, 1, . . . , n − 1), the set nA ± mA has
density 1. Indeed, since m, n are coprime, the set (mA∓i)/n is infinite (this follows
already from the thickness of A). Then, by Lemma 4.9, the set A ± (mA ∓ i)/n,
which we can write as (nA ± mA − i)/n, has density 1. Hence nA ± mA has density
1 in the residue class of i.
Corollary 4.18.
ia + jb = kc.
10
For any i, j, k ∈ N there are a, b, c ∈ A solving the equation
Proof. Restricting to nN, where n = LCD(i, j, k), we can assume that some two
coefficients, for example i and j, are coprime (the other cases can be treated similarly). By Theorem 4.17, iA+jA has relative density 1 in nN, while kA has positive
relative density in nN, so (iA + jA) ∩ kA 6= ∅.
Actually, on has a more general fact:
Theorem 4.19. Let A, B, C be subsets of N and assume that A is classical normal,
B is divisible (contains multiples of every natural number n; note that then B/n is
infinite for each n), and C is substantially divisible, i.e., C/n has positive upper
density for every n. Fix any i, j, k ∈ N. Then the equation ia + jb = kc is solvable
with a ∈ A, b ∈ B, c ∈ C.
Remark 4.20. The assumptions are satisfied when A, B, C are classical normal sets
(the special case A = B = C was treated in Corrolary 4.18).
Proof of Theorem 4.19. Clearly, the set kC/i has positive upper density and the
set jB is divisible, hence jB/i is infinite (regardless of how one interprets jB/i – as
j ·B/i or (jB)/i). By Lemma 4.9, (A+jB/i)∩kC/i has positive upper density (the
same as kC/i). Multiplying by i we obtain that (iA + jB) ∩ kC has positive upper
density, in particular is nonempty. So, there exist (many) desired solutions.
4.5. Solvability of certain equations in net-normal sets.
Motivated by the preceding section, let us now turn to the multiplicative semigroup
(N, ×) and multiplicative normality. The analogue of the equation ia+jb = kc reads
ai bj = ck . To see this analogy even better, let us view (N, ×) again as the direct sum
G which is an additive semigroup. Now the multiplicative equation ai bj = ck takes
on the familiar additive form ia + jb = kc. The problem we immediately encounter
in this “infinite-dimensional” semigroup is that if A is multiplicatively normal (even
net-normal) then the set A/n (multiplicatively this is the set {m : mn ∈ A}) need
not be multiplicatively normal. In fact, it can even be empty, because the set nG
(multiplicatively this is the set of nth powers) has universal multiplicative density
zero. Below we provide an easy example of failure for the multiplicative equation
a2 b2 = c3 , regardless of the Følner sequence in (N, ×).
10For a more general result of this type, presented in a different language and with a different
proof see [Fi2, Theorem 1.3.2].
24
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
Example 4.21. Let (Fn ) be a Følner sequence in (N, ×) and let A be an (Fn )-normal
set (alternatively, it can be net-normal). By removing from A all squares (note that
the set of squares is a set of universal multiplicative density 0), we can assume that
A contains no squares. Then the elements c3 with c ∈ A are not squares either.
Thus A contains no solutions of a2 b2 = c3 .
This is why we will restrict to only the case with i = j = 1, i.e., consider only
equations of the form ab = ck .
Theorem 4.22. Let A and B be net-normal sets and let C contain infinitely many
pairs n, rn, where r is fixed, while n tends to infinity with respect to the multiplicative order < (i.e., for k ∈ N large enough n is a multiple of k; this holds, for
instance, if C is (Fn )-normal with respect to some fixed Følner sequence in (N, ×)).
Then for any natural k there exist a ∈ A, b ∈ B and c ∈ C such that ab = ck .
Corollary 4.23. If A is net-normal then, for any natural k, the equation ab = ck
is solvable in A.
Proof of Theorem 4.22. We continue to switch freely between the sets A, B, C and
their indicator functions denoted xA , xB , xC . We view (N, ×) again as the additive
semigroup G. Thus our task becomes finding solutions of the equation a + b = kc
with a ∈ A, b ∈ B, c ∈ C. Fix a small ε > 0. From now on, adjectives “small”,
“nearly”, “close”, etc. will refer to quantities (error terms, distances) that are
estimated above by functions of ε tending to zero as ε → 0.
By assumption, we can find in C two elements n and n + r, where (r ∈ G is fixed
a priori), with n multiplicatively so large that any rectangle F starting at zero and
ending at a number multiplicatively larger than or equal to 3n is (kr, ε)-invariant
and has the property that both A and B have in F a proportion nearly 21 (it is here
that we are using universal nice normality of A and B).
Now suppose that there are no triples a ∈ A, b ∈ B, c ∈ C satisfying a + b = kc.
This implies that within the rectangle F ending at kn A is disjoint from kn − B.
Since the proportion of both sets in F is nearly 12 , these two sets are in fact nearly
complementary within F , i.e., we can write xA (m) = 1 ⇐⇒ xB (kn − m) = 0 and
this will be true except for a small percentage of m’s in F . The same holds with
n + r replacing n within the rectangle ending at k(n + r), in particular, also in F
(because by (kr, ε)-invariance, F misses only a small percentage of the other, larger,
rectangle). This implies that the configuration of symbols in xA (and also in xB )
within F is nearly invariant under the shift by kr, i.e., in most places m ∈ F the
symbols at m and m + kr are the same. This contradicts universal nice normality
of xA (and likewise of xB ): if F is large enough then the proportion of pairs of
identical symbols at positions m and m + kr with m ∈ F should be close to 21 , not
to 1.
Example 4.24. Using an idea similar to that utilized in the proof of Theorem 4.6,
we will show that assuming multiplicative normality with respect to just one nice
Følner sequence may be insufficient for the solvability of the equation ab = c3 . We
continue to use the additive notation of G. Let Ln be a multiplicatively increasing
to infinity sequence of natural numbers. We assume that Ln+1 < 5Ln (recall that
multiplicatively this means L5n |Ln+1 ). Let Fn be the rectangle with the leading
element
S 3Ln and let Bn be Fn with the rectangle ending at 2Ln removed. Let
B = n Bn . Note that as soon as Ln is high-dimensional, say of a large dimension
NORMAL SETS IN AMENABLE SEMIGROUPS
25
d (multiplicatively, this means that Ln is a product of [powers of] d different primes)
then Bn constitutes the large fraction 1 − ( 32 )d of Fn . It is now obvious that B has
(Fn )-density 1. Consider the sum a + b of two elements of B. Let n be the maximal
index such that Bn contains either a or b. Then a+b belongs to the rectangle ending
at 6Ln with the rectangle ending at 2Ln removed (call this difference CS
n ). It is
easy to see (it suffices to consider the one-dimension case) that the union n Cn is
disjoint from 3B. We have shown that a + b = 3c has no solutions in B. Since B
has (Fn )-density 1, it now suffices to intersect it with any (Fn )-normal set to get
an (Fn )-normal set without the considered solutions.
It is now natural to ask: are all the multiplicative equations ab = ck solvable in
classical normal sets? Here the answer is known to be negative. A. Fish constructed
normal sets which are with the help of a multiplicative function (so-called random
Liouville function) f : N → {−1, 1}, as A = {n : f (n) = −1} (see [Fi1]). In such a
set there are clearly no solutions of the equations ab = ck for an odd k.
4.6. Pairs {a + b, ab} in classical additively normal sets. It this subsection we
establish yet another nontrivial property of classical normal sets.
Theorem 4.25. Let A be a classical normal set. For given a ∈ N define
Sa = {b : a + b ∈ A, ab ∈ A}.
Then for every a ∈ N either Sa or Sa2 has positive upper density. In particular, A
contains pairs {a + b, ab} with arbitrarily large a and b.
Remark 4.26. The property stipulated in Theorem 4.25 does not necessarily hold
for general additively normal sets. Indeed, one can construct an additively thick
set which does not contain pairs {a + b, ab} [BM, Theorem 6.2]. Clearly, such a
thick set contains an additively normal set with no pairs {a + b, ab}.
Proof of Theorem 4.25. It follows from the definition of the set A/n that b ∈
Sa ⇐⇒ b ∈ A/a ∩ (A − a). Fix some a ≥ 2 and suppose that both Sa and
Sa2 have density zero. This can be written as
A ∩ (A/a + a) ≈ ∅ and A ∩ (A/a2 + a2 ) ≈ ∅,
where ≈ means equality up to a set of density zero. Since every set in the above
intersections has density 21 , we get A/a + a ≈ N \ A and A/a2 + a2 ≈ N \ A, and in
particular
A/a + a ≈ A/a2 + a2 .
Multiplying both sides by a we obtain
(A ∩ aN) + a2 ≈ (A/a ∩ aN) + a3 .
Since A/a and A − a are nearly disjoint (the intersection has zero density), we also
have that (A/a ∩ aN) + a3 is nearly disjoint from (A ∩ aN) + a3 − a. Plugging this
into the last displayed formula we conclude that (A ∩ aN) is nearly disjoint from
(A ∩ aN) + a3 − a2 − a. Dividing both sets by a, we get that A/a is nearly disjoint
from A/a + a2 − a − 1. We have proved that the indicator function of A/a has the
property that for n’s of density 1 its values at n and at n + r (where r = a2 − a − 1)
are different. This contradicts Lemma 4.15 (normality of A/a), as in normal sets
the density of such n’s should be 21 .
26
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
4.7. Multiplicative configurations in classical normal sets. In this subsection we show that every classical normal set contains (up to scaling) all configurations which are known to be present in multiplicatively large sets. The following
theorem holds the key:
Theorem 4.27. Let A ⊂ N be a classical normal set. Then, for any Følner
sequence (Kn ) in (N, ×) there exists a set E of (Kn )-density 12 such that for any
finite subset {n1 , n2 , . . . , nk } ⊂ E the intersection A/n1 ∩ A/n2 ∩ · · · ∩ A/nk has
positive upper density in (N, +).
The key role in the proof of Theorem 4.27 will be played by the following theorem, the proof of which is based on the proofs of [Be2, Theorem 4.19] and [Be1,
Theorem 2.1].
Theorem 4.28. Let (Fn ) be a Følner sequence in (N, +), let a ∈ (0, 1), and let
F = {A1 , A2 , . . . } be a countable family of subsets in N such that d(Fn ) (A) ≥ a for
all A ∈ F. Then there exists an invariant mean L on the space BC (N) of bounded
complex-valued functions such that
(i) L(1A ) = d(Fn ) (A) for every A ∈ F,
(ii) for any k ∈ N and any n1 , n2 , . . . , nk ∈ N,
d(Fn ) (An1 ∩ An2 ∩ · · · ∩ Ank ) ≥ L(1An1 · 1An2 · . . . · 1Ank ),
(iii) there exists a compact metric space X, a regular measure µ on B(X) (the
Borel σ-algebra of X), and sets Ãn ∈ B(X), n ∈ N, such that for any
n1 , n2 , . . . , nk ∈ N one has11
L(An1 ∩ An2 ∩ · · · ∩ Ank ) = µ(Ãn1 ∩ Ãn2 ∩ · · · ∩ Ãnk ).
Proof. Let S be the (countable) family of all finite intersections of the form An1 ∩
An2 ∩ · · · ∩ Ank , where Anj ∈ F, j = 1, . . . , k. By using the diagonal procedure we
arrive at a subsequence (Fni ) of our Følner sequence (Fn ), such that for any S ∈ S
the limit
X
1
|S ∩ Fni |
= lim
1S (m)
L(S) = lim
i→∞ |Fni |
i→∞
|Fni |
m∈Fni
exists. Notice that L(A) = d(Fn ) (A) for any A ∈ F, and that for any n1 , n2 , . . . , nk ∈
N we have
T


k
k
j=1 Anj ∩ Fn \
d(Fn ) 
≥
Anj  = lim sup
|Fn |
n→∞
j=1
T


k
k
j=1 Anj ∩ Fni \
lim
= L
An j  .
i→∞
|Fni |
j=1
Extending by linearity, we get a linear functional L on a subspace V ⊂ BR (N).
By invoking the Hahn-Banach Theorem, we can extend L from V to BR (N). This
L naturally extends to a functional on the space BC (N), which satisfies conditions
(i) and (ii).
We move now to proving (iii). Let A be the uniformly closed and closed under
conjugation algebra of functions on N, which is generated by indicator functions 1A
11We view, when convenient, L as a finitely additive measure on P(N).
NORMAL SETS IN AMENABLE SEMIGROUPS
27
of sets A ∈ F. Then A is a separable C ∗ -subalgebra of `∞ (N, k · k∞ ), and, by the
Gelfand Representation Theorem, A ∼
= C(X), where X is a compact metric space.
The restriction LA of the mean L, which we constructed above, induces a positive
linear functional L̃ on C(X), which by the Riesz Representation Theorem is given
by a Borel measure µ.12
Note that the isomorphism A ∼
= C(X) sends indicator functions of subsets of N
to indicator functions of subsets of X (because the isomorphism provided by the
Gelfand transform preserves algebraic operations, and the indicator functions are
the only ones which satisfy the equation f 2 = f ). Let Ãj be the subsets of X which
correspond to sets Aj ∈ F (note that since 1Ãj ∈ C(X) for each j, the sets Ãj are
measurable). Clearly, we have
LA (An1 ∩ An2 ∩ · · · ∩ Ank ) = µ(Ãn1 ∩ Ãn2 ∩ · · · ∩ Ãnk )
for any n1 , n2 , . . . , nk . This completes the proof.
The last result which is needed for the proof of Theorem 4.27, is the following
theorem.
Theorem 4.29 (see Lemma 5.10 in [Be4]). Let (Kn ) be a Følner sequence in
(N, ×), let (X, B, µ) be a probability space, and let Aj , j ∈ N be measurable sets in
X, satisfying µ(Aj ) ≥ a > 0 for every j ∈ N. Then there exists
T a set E ∈ N with
d(Kn ) (E) ≥ a, such that for any finite set F ⊂ E, one has µ
j∈F Aj > 0.
Proof of Theorem 4.27. The result in question follows from Theorem 4.28 applied
to the Følner sequence Fn = {1, 2, . . . , n} in (N, +) and F = {A/n : n ∈ N}. Note
that, by Lemma 4.15, each A/n is a classical normal set and hence has density
a = 21 .
Corollary 4.30. Let S M denote the family of all finite sets in N which have the
following property:
• any multiplicatively large set in N contains a configuration of the form bF ,
where b ∈ N, F ∈ S M .
Then for any F ∈ S M every classical normal set A contains bF for some b ∈ N.
Proof. Let F = {n1 , n2 , . . . , nk } ∈ S M . The mulitplicatively large set E in the
formulation of Theorem 4.27 contains bF for some b ∈ N. Theorem 4.27 implies
that A/bn1 ∩ A/bn2 ∩ · · · ∩ A/bnk 6= ∅. Any element c in this intersection satisfies
cbF ⊂ A.
It was shown in [Be3] that multiplicatively large sets in N have very rich combinatorial structure (which is quite a bit richer than that of additively large sets). For
example, any multiplicatively large set contains not only arbitrarily long geometric
and arithmetic progressions, but also all kinds of more complex structures which
involve both the addition and multiplication operations. Corollary 4.30 allows to
conclude that classical normal sets in N are, in a way, as combinarorially rich as
multiplicatively large sets. For example, [Be3, Theorem 3.10] gives the following
result.
12We remark that for our applications we need only a “restricted” version of Theorem 4.28
which deals with functional LA on A and does not need appealing to the Hahn–Banach Theorem.
28
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
Theorem 4.31. Let S a and S m be two families of finite sets in N which have the
following properties:
(i) any additively large set in N contains a configuration of the form a+F1 , where
a ∈ N, F1 ∈ S a ,
(ii) any multiplicatively large set in N contains a configuration of the form bF2 ,
where b ∈ N, F2 ∈ S m .
Then any classical normal set A ⊂ N contains a configuration of the form
bF2 (a + F1 ) with a, b ∈ N, F1 ∈ S a , F2 ∈ S m . In particular, any classical normal set A contains arbitrarily large sets of the form {q j (a + id); 0 ≤ i, j ≤ n}.
Similarly, invoking [Be3, Theorem 3.15] one gets the following:
Theorem 4.32. Any classical normal set contains arbitrarily large configurations
of the form {b(a + id)j ; 0 ≤ i, j ≤ n}.
Remark 4.33. Note that Corollary 4.30, and thus Theorems 4.31 and 4.32, are not
valid for general additively normal sets in (N, +). For example, one can show (see
[BBHS, Theorem 3.5]) that there exist additively thick sets which do not contain
geometric progressions of length 3, {c, cr, cr2 }, where r ∈ Q\{1} (cf. Remark 4.26).
5. Multiplicatively normal Liouville numbers
Let us recall that x ∈ R is called a Liouville number if for every natural k there
exists a rational number pq such that |x − pq | < q1k . Clearly, “every k” can be
equivalently replaced by “arbitrarily large k” (if pq is good for k, it is also good for
all k 0 < k). It is well known that the set of Liouville numbers is residual (dense
Gδ ) but its Lebesgue measure equals zero.
Fact 5.1. Some Liouville numbers can be obtained in the following inductive manner: Take any binary block w2 , then repeat it two times. Continue by appending
any block C2 . That is, we create a block w3 = w2 w2 v2 . Then repeat w3 three times
and append any v3 , i.e., define w4 = w3 w3 w3 v3 . And so on. In the end (as the
limit of the blocks wk ) we obtain a binary expansion of a Liouville number x.
Proof. Given k, consider the rational number pq represented by the periodic sequence wk wk . . . . The denominator q is less than 2|wk | , hence q1k > 2−k|wk | . The
difference |x − pq | is a number whose first nonzero digit appears at a position larger
than k|wk |, which means that it is at most 2−k|wk | , hence less than q1k .
In this section we will prove the following fact:
Theorem 5.2. For any nice Følner sequence (Fn ) in (N, ×) there exists an (Fn )normal Liouville number.
We remind that M < m is synonomous with m|M and if additionally M 6= m, we
will write M m. We also remind that a nice Følner sequence (Fn ) can be identified
with a multiplicatively increasing sequence (Ln ) (i.e., such that Ln+1 Ln for
each n) of leading elements (then Fn = {m : Ln < m}).
We begin the proof with a lemma:
Lemma 5.3. Given k ≥ 2 and ε > 0, there exists an mk,ε such that for any
m < mk,ε and any M < m, the interval [m, km) contains at most a fraction ε of
all divisors of M .
NORMAL SETS IN AMENABLE SEMIGROUPS
29
Proof. Let p be the smallest prime number larger than or equal to k. Let r ∈ N
be such that 1r ≤ ε, and put mk,ε = pr . Let m be any multiple of mk,ε and
let M be any multiple of m. The set of all divisors of M (i.e., the anchored
rectangle ending at M ) splits into disjoint union of one-dimensional sets of the
form aI = {a, ap, ap2 , . . . , aps }, where a is not a multiple of p, and ps is the largest
power of p dividing M . Clearly, s ≥ r. Since p ≥ k, at most one element from any
set aI can fall in [m, km). Thus at most the fraction 1r ≤ ε of all divisors of M fall
in [m, km).
Proof of Theorem 5.2. The proof consists in choosing an arbitrary (Fn )-normal
number x0 and then modifying its binary expansion on a set of (Fn )-density zero.
Of course, any such modification preserves (Fn )-normality. On the other hand, the
modification will lead to a binary expansion which fits the description in Fact 5.1.
For each k abbreviate mk,2−k from the preceding lemma as mk . Each of these
numbers can be replaced by any of its multiples, in particular, by LCM(mk , Ln−1 ),
where n is the smallest index such that mk ∈ Fn . In this manner, the (new)
numbers mk satisfy for each n either mk Ln or Ln < mk . Also, the role of mk
can be played by mk0 for any k 0 ≥ k, hence passing, if necessary, to a subsequence
we can assume that mk+1 > kmk , for each k.
Now we start modifying the binary expansion of the (Fn )-normal number x0 , as
follows. We keep the entries in [1, m2 ) unchanged. This is our block w2 . Next, we
replace the symbols along [m2 , 2m2 ) by a repetition of w2 . We leave the entries in
[2m2 , m3 ) unchanged (this is our block v2 ). The modified sequence now starts with
w3 = w2 w2 v2 . Then we replace the symbols along [m3 , 3m3 ) by w3 w3 and we leave
the entries along [3m3 , m4 ) unaltered. The rest of the inductive procedure should
now be clear. According to Fact 5.1, the modification leads to a Liouville number
x.
S It remains to compute the (Fn )-density of the set of modified coordinates,
k≥2 [mk , kmk ). Take a large set (Fn ) (the divisors of Ln ). For very small k the
fraction of Fn that falls in [mk , kmk ) is negligibly small. For larger (but not too
large) k we still have Ln < mk hence, by Lemma 5.3, the corresponding fraction
does not exceed 2−k . Finally, for larger k, mk Ln , in particular mk > Ln , hence
the corresponding fraction is zero. Jointly, the contribution of the above union
in Fn is approximately equal to the sum of a finite geometric progression (2−k )
with some (increasing with n) number of the initial terms ignored. This yields the
desired (Fn )-density zero of the set of modified coordinates.
Remark 5.4. The above technique allows to produce more than just (Fn )-normal
Liouville numbers. If we consider any property satisfied by a nonempty set of numbers and preserved under zero (Fn )-density modifications of the binary expansion,
such as being generic for some multiplicatively invariant (not necessarily Bernoulli)
measure, (Fn )-frequency α ∈ [0, 1] of 1’s, etc., then there exists a Liouville number
with that property (of course, as long as (Fn ) is a nice Følner sequence).
Remark 5.5. The above technique does not allow to produce a Liouville number
which would be net-normal. It is so because the union of any fixed sequence of
intervals of the form [mk , kmk ) has upper density at least 21 for a suitable nice
Følner sequence. To prove this, it suffices to indicate, for any leading element L,
its multiple pL half of whose divisors falls in one of the intervals [mk , kmk ). To
this end choose k larger than 2L and a prime number p in [mk , 2mk ] (p exists by
30
VITALY BERGELSON, TOMASZ DOWNAROWICZ, AND MICHAL MISIUREWICZ
Bertrand–Tchebychev Theorem, see [Tch]). Then at least half of the divisors of pL
have the form pl, where l ≤ L and then mk ≤ p ≤ pl ≤ 2mk L < kmk .
Remark 5.6. On the other hand, the preceding remark does not exclude the existence of a net-normal Liouville number, even one of the form described in Fact 5.1.
Although the union of the “intervals of repetitions” [mk , kmk ) has positive upper
density (for some nice Følner sequences), their periodic contents might still have
good multiplicative normality properties. We leave the existence of such a number
as an open problem.
Acknowledgement. The first author gratefully acknowledges the support of
the NSF under grant DMS-1500575. The research of the second author is supported
by the NCN (National Science Center, Poland) Grant 2013/08/A/ST1/00275. The
work of the third author was supported by the grant number 426602 from the
Simons Foundation.
References
[AKS]
R. Adler, M. Keane and M. Smorodinsky, A construction of a normal number for the
continued fraction transformation, J. Number Theory 13 (1981), 95–105.
[AdJ] M. A. Akcoglu and A. del Junco, Convergence of averages of point transformations, Proc.
Amer. Math. Soc. 49 (1975), 265–266.
[BB]
D. H. Bailey and J. M. Borwein, Normal numbers and pseudorandom generators, in:
“Computational and analytical mathematics,” Springer Proc. Math. Stat. 50, Springer,
New York, 2013, pp. 1–18.
[BBHS] M. Beiglböck, V. Bergelson, N. Hindman and D. Strauss, Multiplicative structures in
additively large sets, J. Combin. Theory Ser. A 113 (2006), 1219–1242.
[Be1]
V. Bergelson, Sets of recurrence of Zm -actions and properties of sets of differences in
Zm , J. London Math. Soc. 31 (1985), 295–304.
[Be2]
V. Bergelson, Ergodic theory and Diophantine problems, in: “Topics in symbolic dynamics
and applications (Temuco, 1997),” London Math. Soc. Lecture Note Ser. 279, Cambridge
Univ. Press, Cambridge, 2000, pp. 167–205.
[Be3]
V. Bergelson, Multiplicatively large sets and ergodic Ramsey theory, Israel J. Math. 148
(2005), 23–40.
[Be4]
V. Bergelson, Combinatorial and Diophantine Applications of Ergodic Theory, in:
“Handbook of Dynamical Systems,” vol. 1B, Elsevier, Amsterdam, 2006, pp. 745–869.
[Be5]
V. Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, in: “Ultrafilters across mathematics,” Contemp. Math. 530, Amer. Math. Soc., Providence, RI,
2010, pp. 23–47.
[BM]
V. Bergelson and J. Moreira, Measure preserving actions of affine semigroups and {x +
y, xy} patterns, Ergod. Th. Dynam. Sys. (online, doi:10.1017/etds.2016.39, 26 pages).
[BW]
V. Bergelson and B. Weiss, Translation properties of sets of positive upper density, Proc.
Amer. Math. Soc. 94 (1985), 371–376.
[Bo]
E. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ.
Mat. Palermo 27 (1909), 247–271.
[Br]
A. A. Brudno, Entropy and the complexity of the trajectories of a dynamic system (Russian), Trudy Moskov. Mat. Obshch. 44 (1982), 124–149.
[Bu]
Y. Bugeaud, Nombres de Liouville et nombres normaux, C. R. Math. Acad. Sci. Paris
335 (2002), 117–120.
[Ch]
D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London
Math. Soc. 8 (1933), 254–260.
[CP]
A. H. Clifford and G. B. Preston, “The algebraic theory of semigroups” Vol. I, Mathematical Surveys, No. 7, Amer. Math. Soc., Providence, RI, 1961.
[DE]
H. Davenport and P. Erdős, Note on normal decimals, Canad. J. Math. 4 (1952), 58–63.
NORMAL SETS IN AMENABLE SEMIGROUPS
[DHZ]
[Fi1]
[Fi2]
[Fu1]
[Fu2]
[GRS]
[Ha]
[Ko]
[La]
[Li]
[vM]
[PV]
[OW]
[Tch]
[Wa]
31
T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups,
Journal für die reine und angewandte Mathematik (Crelles Journal), online
https://doi.org/10.1515/crelle-2016-0025
A. Fish, Random Liouville functions and normal sets, Acta Arithmetica 120 (2005),
191–196.
A. Fish, Solvability of linear equations within weak mixing sets, Israel J. Math. 184
(2011), 477–504.
H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1–49.
H. Furstenberg, “Recurrence in ergodic theory and combinatorial number theory,” M. B.
Porter Lectures, Princeton University Press, Princeton, N.J., 1981.
R. L. Graham, B. L. Rothschild and J. H. Spencer, “Ramsey theory,” Second ed., John
Wiley & Sons, Inc., New York, 1990.
H. A. Hanson, Some relations between various types of normality of numbers, Canad. J.
Math. 6 (1954), 477–485.
J.-M. De Koninck, The mysterious world of normal numbers, in: “Scalable uncertainty
management,” Lecture Notes in Comput. Sci., 9310, Lecture Notes in Artificial Intelligence, Springer, Cham, 2015, pp. 3–18.
J. Lambek, The immersibility of a semigroup into a group, Canadian J. Math. 3 (1951),
34–43.
E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001),
259–295.
R. von Mises, Über Zahlenfolgen, die ein kollektiv-ähnliches Verhalten zeigen, Math.
Ann. 108 (1933), 757–772.
P. Pollack and J. Vandehey, Some normal numbers generated by arithmetic functions,
Canad. Math. Bull. 58 (2015), 160–173.
D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable
groups, J. Anal. Math. 48 (1987), 1–141.
P. Tchebychev, Mémoire sur les nombres premiers. Journal de Mathématiques pures et
appliquées, Sér. 1 (1852), 366–390.
D. D. Wall, Normal numbers, Thesis (Ph.D.), University of California, Berkeley, 1950.
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
E-mail address: [email protected]
Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 21, 50-370 Wroclaw, Poland
E-mail address: [email protected]
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202, USA
E-mail address: [email protected]