ON IRREGULARITIES OF DISTRIBUTION
W. W. L. CHEN
§1. Introduction. Let Uo = [0, 1) and Ul = (0,1]. Suppose we have a
distribution & of N points in Uk0+1, where, for k ^ l,Uk0+1 is the unit cube
with 0 ^ y, < 1 (i = 1,...,fc+ 1). For
consisting of the points y = (yi,-.,yk+1)
x = (xj,..., xk+1) in U\+1, let B{x) denote the box consisting of all y such that
0 < yt < xt (i = 1,..., k+1), and let Z\0>; B(x)] denote the number of points of &
which lie in B(\). Write
The irregularity of the distribution 3P can be measured in a number of ways by the
behaviour of the function D[& ; B(x)]. One may consider the iT-norm
\\D(n\w = (
\D[<?;B(x)Tdx1...dxk+1
...
u,
y.
Roth [3] obtained a lower bound for the L2-norm, and Schmidt [7] established
the following generalization, Roth's result being the special case W = 2.
THEOREM 1. (Schmidt [7]). For every W> 1, there exists a positive number
cy{k, W), depending only on k and W, such that
\\D(P)\\W>
k
Cl(k,W)(\ogN)-
.
Roth's lower bound for the L2-norm has been shown to be sharp, apart from the
value of the constant. This was established in the special cases k = 1 and k = 2 by
Davenport [1] and Roth [5] respectively, and more recently for arbitrary k by
Roth [6]. Different proofs for the case k = 1 were given by Vilenkin [8], Halton and
Zaremba [2] and Roth [4].
The object of this paper is to show that Schmidt's lower bound, given in Theorem
1 above, is also sharp, apart from the value of the constant. Accordingly, we shall
prove the following theorem.
THEOREM 2. Let W > 0. For a suitable number c2(k, W), depending only on k and
W, there exists, corresponding to every natural number N ^ 2, a distribution 3P, which
may depend on W, of N points in Uk0+1 such that
\\D(n\w<c2(k,W)(\ogN)ik.
The two-dimensional case, fc = 1, of Theorem 2 can easily be deduced from the
argument in Roth [4]. On the other hand, since the L^-norm increases with W, the
[MATHEMATIKA, 27 (1980), 153-170]
154
W. W. L. CHEN
case W ^ 2 of Theorem 2 follows immediately as a consequence of Roth [6]. Ou:
result is therefore new only when k ^ - 2 , W> 2.
Our method is, as in Roth [6], based on the consideration of "translations" of the
Hammersley (Halton) sequence, an average with respect to such translations being
taken over a "long" interval. In the application of the Main Lemma to deduce
Theorem 2 (see §7), the set Q = Q (1) is chosen in terms of the Hammersley (Halton)
sequence^ exactly as in [6]. With this choice of Q, Roth's "Basic Lemma" in [6] is
the case W = 2 of our Main Lemma. However, to prove the Main Lemma, new ideas
are required to overcome difficulties that did not arise in the case W = 2. The prooi
of the Main Lemma is new, and is based on the use of induction which was not
needed in the special case W = 2 dealt with by Roth in [6]. To make the induction
work, we have to make use of sequences (sets) (see §3) which are of a more general
nature than the Hammersley (Halton) sequence, whilst retaining its relevant
properties. We therefore have to introduce a wider class of sets Q to give a
formulation of our Main Lemma suitable for proof by induction. We also remark
that the derivation of Lemma 5, a key identity in the proof of the Main Lemma,
requires an elaboration of the idea underlying Roth's proof of his Basic Lemma (see
(24) and (36)).
I am greatly indebted to Professor Roth for his constant encouragement and
helpful suggestions, and for giving me manuscripts of his papers [5] and [6]. I alsc
thank the referee for his valuable comments.
§2. Notation. By an interval /, we shall mean a half-open interval of the type
[a 1 ; a 2 ), while It x I2 denotes the Cartesian product of 7X and / 2 .
We use R to denote a residue class. In particular, R(m, q) denotes the residue
class of integers congruent to m modulo q. For any real number t, we denote by t + B
the set {t + n : n e R}, and write
where Z[t + i ? ; / ] denotes the number of elements of t + R falling into /, q is the
modulus of the residue class R, and <?(/) is the length of / . It is obvious that
/]|<l
always.
Throughout, Z denotes the set of all integers.
(2
§3. Generalization of the idea of the Hammersley (Halton) sequence. Let h be £
positive integer, and let py,...,pk
be distinct primes. F o r any integer n satisfying
0 ^ n < (p! ... pk)h and for every j = 1,..., k, write
00
v=0
where £ ' signifies that there is only a finite number of non-zero terms in the sum. Ii
t In fact, any choice of Q (1) which satisfies the Main Lemma would do. However, the value of t* in (73
ll)
would then depend on our choice
ON IRREGULARITIES OF DISTRIBUTION
155
is clear that the integers ajv are uniquely determined by n. We write
00
Xj(n) = p / 1 X <*j.,Pr •
v = 0
Then Xj(n) lies in Uo. Let
X(n) = (*i(«)>-,**(«))•
Then the vectors
are the first (pj ... p,,)'1 terms of the Hammersley (Halton) sequence. We extend the
range of definition of X(«) over the set Z of all integers by periodicity so as to ensure
that
X(n + (Pl...Pk)h)
=
for every integer n. In [6], Roth considered the set
neZ},
(3)
where (X(n), n + t) stands for the point (Xr(n),..., Xk(n), n + t) in (k+ l)-dimensional
space.
In this paper, we need to consider sets which are of a more general nature than
(3). Let h be a positive integer, and let p1,..., pk be distinct primes. We define a q-set
of class h with respect to the primes p1,...,pk.
Let 1 ^ j ^ k.
Definition. Let 0 ^ s ^ h. An interval [ a 1 ; a 2 ) is said to be an elementary
p^type interval of order s if a 1 ) a 2 are consecutive integer multiples of p^s, and
[a 1( <x2) is contained in Uo.
We reserve the symbol J for elementary intervals.
By an association of order h with respect to the prime p j ; we shall mean the
following. We associate each elementary p^-type interval of order s with a residue class
modulo pSj in such a way that if Jt and J2 are elementary p^-type intervals, of orders
sx and s2 respectively, where 0 ^ s2 ^ s1 ^ h, and if Rx and R2 are respectively their
associated residue classes, then
JlCJ2
o
R^R2.
(4)
Since either Jt <= J2 or Jt n J2 =0, and a similar relation applies to Rt and R2, we
see, in particular, that this association sets up a one-to-one correspondence between
the collection of elementary Pj-type intervals of order s and the collection of residue
classes modulo p) for every s satisfying 0 < s ^ h.
Let Q be a residue class modulo q, where (q, px ... pk) = 1, and suppose
SC = {x(n): n e Q} is a set, where for each n e g , x(«) = (x^n),..., xk(n)) is a point in
UQ. Consider the set
We are concerned with sets of the form (5) and having a very special property.
Our next definition is motivated by a special property of O'(0), where fi'(t) is the
set (3).
156
W. W. L. CHEN
Definition. Suppose 3C is such that, for every integer j = 1,..., k, we can define
an association between elementary p^-type intervals and residue classes modulo pj
(0 ^ s ^ h)'as follows. Given any residue class R modulo p), where 0 < s ^ h, there
exists an elementary Pj-type interval J of order s such that {x/n): n e Q n R} c J; in
other words, the 7-th coordinates of those points x(n), for which n belongs to a
residue class modulo p), as well as Q, all fall into the same elementary p^-type interval
of order s. We associate J with R. Suppose further that, for every j = l,...,/c, the
corresponding association is of order h with respect to the prime py, in other words,
that for every; = 1,..., k, the set 9C sets up an association of order h with respect to
the prime pj. Then we say that Q(Q, 'X) is a q-set of class h with respect to the
primes
p1,...,pk.
We note that, given any q-set il(Q, 9C) of class h with respect to the primes
p1,...,pk,
the association of order h with respect to each prime is uniquely
determined.
For the sake of simplicity, since it is important only to bear in mind the modulus
q of the residue class Q when considering sets of the type O(g, 3C\ we write Q<q) for
Q(g, 3C). We shall consider sets of the form
neQ\.
(6)
Note that Q(4)(0) is the set (5).
Definition. Suppose Q{q) is a q-set of class h with respect to the primes pt,..., pk;
and suppose t is any real number. We say that the set Q(<!)(t) is a translated q-set of
class h with respect to the same primes.
It is clear that the set (3) given by the Hammersley (Halton) sequence is a
translated 1-set of class h with respect to the primes px,..., pk. Furthermore, if Q is a
residue class modulo q, where (q,px ...pk) = 1, the subset {(X(n), n + t): ne Q} of
(3) is a translated q-set of class h with respect to the primes p t ,..-., pk. On the other
hand, the following lemmas are obvious.
1. Suppose Q(?)(t) is a translated q-set of class h with respect to the primes
pl,...,pk.
Then, for every s satisfying 1 < s ^ h,Cl(q){t) is also a translated q-set of
class s with respect to the same primes.
LEMMA
LEMMA 2. Suppose fi(1)(t) = {(x(n), n + t):neZ}
is a translated 1-set of class h
with respect to the primes p 1 ; . . . , pk. Then, for every j = 1,..., k, the set
{(x^n),...,
Xj-^n), xj+l(n),...,
xk(n), n +
t):neZ}
is a translated 1-set of class h with respect to the primes px,..., Pj-X, Pj + l, ••-,?*• On
the other hand, if Ro is a residue class modulo q, where (q, pt ... pk) = 1, then the set
{(x(n), n + t):ne Ro} is a translated q-set with respect to the primes p l 5 . . . , pk.
Suppose B = Ix x... x Ik x /*, where Ij c l/0 (1 < j ^ k) and /* is of the type
[0, Y), where Y is positive but otherwise unrestricted. For any Q(q)(t) with respect to
the primes p 1 ? ..., pk, we define £[Q ( 9 ) (t); B~\ by
»(t); B] = Z[Q<«>[t);B\-q-lV{B),
(7)
ON IRREGULARITIES OF DISTRIBUTION
157
where Z[Qiq)(t); B] is the number of points of Qiq)(t) falling into B, and V(B) is the
volume of B. Note that q~l V{B) is the "expected number" of points of Q(9)(t) falling
into B. The analogues of (7) in lower-dimensional cases are defined in the same way.
It is clear that if B = BY u B2, where BlnB2
= 0, then
£[n<«>(r); B] = E[Q(«>(t); B^ + El^q\t);
B2] .
(8)
Definition. Let
0 < s < h.
Suppose
further
that,
for
each
j = l,...,k,Ij
= [0, r/j), where 0 < n},^ 1 and ns is an integer multiple of pjs. Then
we say that B is a box of class s with respect to the primes
p1,...,pk.
We reserve the symbol B* for such boxes.
3. Suppose Q(9) is a q-set of class h with respect to the primes pY,..., pk,
where (q,px •••pk) = 1," and suppose B* is a box of class s~with respect to the same
primes, where 0 < s < h. Then the function E[C^q)(t); 5*] is periodic in t with period
LEMMA
Proof. Since B* is a box of class s with respect to the primes px,..., pk, B* is a
finite and mutually disjoint union of boxes of the type B = J j x ... xjkxl*,
where,
for; = 1,..., k, JjAs an elementary p r type interval of order s. By (8), it suffices to
prove Lemma 3 with B* replaced by such a box B. Consider, for any j = 1,..., k, the
set Qj = {n e Q : Xj(n) e Jj}. Then Qi, = Q n Rj, where Rj is the associated residue
class (modulo p)) of Jj. By the Chinese Remainder Theorem, the set
R1 n ... n Rk, = R say, is a residue class modulo (pt ... pkf. Since (q, pl ... pk) = 1,
we see that Q n R is a residue class modulo ^(pj ... p t ) s . It follows from (1), (6) and
(7) that for the above B,
E[&q)
(t) • B ] = F[t + (Q n / ? ) ; / * ] .
The proof of Lemma 3 is now complete, since F[t + (Q n i?); /*] is clearly periodic
with the desired period.
§4. Statement of the basic result and an important identity. In this section and
the next two, we establish the following result from which Theorem 2 will be deduced
in §7. We also assume that W is an even positive integer. Let
M = Pl...Pk;
(9)
and, for j = 1,..., k, write
Mj = p1...pj_ipj+1...pk
= MpJ1 .
(10)
MAIN LEMMA. For a suitable constant C(p1,..., pk, W), depending only on W and
the primes p 1 ; ...,pk, we have, for any l-set Q (1) of class h with respect to the primes
p t , ...,pk, and, for any box B* of class h with respect to the same primes,
M>>
[
C{Pl,...,pk,W)Mhh*kw.
(11)
158
W. W. L. CHEN
The Main Lemma is in fact equivalent to the following superficially more general
form.
ALTERNATIVE FORM OF THE MAIN LEMMA. For the same constant C(py,..., pk, W)
as in the Main Lemma, for any q-set Qiq) of class h with respect to the primes p l 5 ..., pk,
where (q, p1 ... pk) = 1, and for any box B* of class h with respect to the same primes,
we have
qMh
I E[_niq)(t);B*]wdt s? C(Pl,...,pk,
W)qMhhikw.
(12)
o
To see this equivalence, we first note that if Q{q)(t) = O(Q, 3C; t), where Q is the
residue class n0 modulo q, then, by Lemma 3, the left-hand side of (12) is equal to
q
E[_Q.(Q,%;qt'-no);B*Ydt',
where
Q{Q, 3C ; qt'— n0) = {(x(n), n — no + qt'): ne Q}.
We
now
Q(Z, T ; t') = {(x'(n'), «' +1') -ri eZ}
by
writing,
for
each ri
x'(n') = x(no + qri), so that
{{x(n\q-l(n-n0)
+ t'):neQ}
= {{x\ri),ri+ t'):ri sZ] .
define
e Z,
(13)
Let
(B1)* be
derived
from
B* = IY x... x Ikx [0, Y)
by
writing
(F)* = / 1 x . . . x / l x [ 0 , f 1 y ) . It now suffices to show that Q(Z,^';t') is a
translated 1-set of class h with respect to the primes px,..., pk, so that by (7) and (13),
E[Q{Q, 3C ; qf-n0);
B*] = Z[Q{Q, 9C ; qf-n0);
B*1-q'lV(B*
= Z[Q(Z, T • t'); (B')*] -
(
)
and our argument is complete. It now remains to show that, for any j = 1,..., k, the
set #"' = {x'(n'): n' e Z} sets up an association of order h with respect to the prime
Pj. In other words, we have to show that for any j = 1,..., k and for any s satisfying
0 ^ s < h, the set of those integers ri for which x'j(ri) falls into any fixed elementary
Py-type interval J of order s constitutes a residue class modulo pj. Since Q(6, 3C) is a
q-set of class h with respect to the primes p1,...,pk, the set of those n e Q for which
Xj(n) falls into J is contained in a residue class R modulo p). Then it is clear that
{ri : x'jiri) e J} = {n': n0 + qn' e g A Xj{n0 + qri) e J}
= {ri :no + qri eQ n /?},
where Q n R is a residue class modulo qpj. Furthermore,
gnR
<=> ri e R',
ON IRREGULARITIES OF DISTRIBUTION
159
where R' is some residue class modulo psj.
To prove the Main Lemma, we induce on h. For k ^ 2, we also make use of the
corresponding results in a lower-dimensional case. We therefore begin by
investigating some properties of Q U) and B*.
Let SC = {x(n): n e Z} be any arbitrary set such that the set Q. = Q(1) defined by
(5) is a 1-set of class h with respect to the primes p1,..., pk. We keep this set SC fixed
throughout.
Suppose
B* = B*(t,, Y) = [0, ni) x ... x [0, r,k) x [0, Y)
(14)
is a box of class h with respect to the primes pi,...,pk.
For j = l,...,fe and
s = 0, ...,/i, let £JtS denote the greatest integer multiple of p ^ s not exceeding rjy,
furthermore, for s =/= 0, write
Then vJS is an integer and 0 =$ vjiS < pj for j = 1,..., k and s = 1,..., h. Let
/* = [0, Y).
(15)
Bs* = [ 0 , £ 1 , s ) x . . . x [ 0 , £ t , s ) x / * ,
(16)
For s = 0,...,fr, let
so that B* is the largest box of class s with respect to the primes p x ,..., pk which is
contained in B*. We now consider the complement of Bf^1 in B*. We let Bx s denote
the part of this which is contained in [ ^ ^ - i , £liS) x [0,1)*1"1 x /*, B2 s denote the
part of the remainder which is contained in [0,1) x [<^2,s-i> £i,s)x [0. If 2 x /*, and
so on. In other words, for j = 1,..., k and s = 1,..., /i, we let
Bj.. = [0, «!.,-!) X ... X [0, ^ ^ . ^ J X [^. , _ , , ^,.)
x[(UJ.+ 1 , s ) x . . . x [ ( U M ) x / * .
(17)
Then, it follows that, for s = 1,..., h,
B* = B * _ 1 u B 1 , , u . . . u B l k , , >
(18)
and that the union is pairwise disjoint; so it follows from (8) that, for s = 1,..., h,
E[Cl{t); Bs*] = £[Q(t); B*_ J + £ £[Q(t); B,,J .
(19)
If, for some 7, s with 1 ^ j ^ k and 1 < s < h, Bjs — 0, then obviously
£[H(t); Bjs~] = 0, and we may omit reference to £[Q(t);B J S ] in (19). So we may
suppose that all BJs are non-empty. Then the interval \_£,hs-x, £,-,s) in (17) is a union
of exactly v,-jS disjoint elementary p,-type intervals of order s. We denote them by
Jj.s.z (a = 1' •••> vj,sX a n d, for future reference, we suppose they correspond
10U
W. W. L. CHEN
respectively to the residue classes R(m(j, s, a), pf). For a = 1,..., vjs, we write
x TO £
)x
x TO £ ) x /*
(20)
Then we see that, for j = 1,..., k and s = 1,..., h,
B
j,s = U
B
j,s,«'
and the union is pairwise disjoint, so that by (8),
E[Q(t); BjJ = £ £[fi(t); Bf,,,, J .
(21)
a= 1
For 7 = 1,..., k and s = 1,...,ft— 1, let
7/,. = K j , . . ^ . . + P7').
(22)
so that J, s is the elementary p^-type interval of order s containing the complement of
[0, <^s) in [0, rij). We modify Bhh (see (17) with s = h) by replacing [ ^ - i , ^, h ) by
J J;S , which is the elementary p^-type interval of order s containing [^ v ,_ 1 ,^ jfc ).
Accordingly, for j = l,...,k and s = l , . . . , n - l , we let the modification Bjs be
defined by
x [0, £•
) x . . x [0 £ ) xI* .
(23)
Note that Bjh hasft— 1 modifications, namely Bj
l,...,Bjh_1.
Our next lemma contains the key idea underlying Roth's proof of his Basic
Lemma in [6].
LEMMA
4.
For j = 1,..., k and for a = 1,..., vjh, we have
Also, for j = 1,..., k and s = 2,..., ft— 1, we have
Py-l
^ £[Q(t + aM)p5" 1 ); BJjS] = E[Q(t); B;, s _ x ] .
(25)
Proof We first prove (24). Recall the definitions of B,._M (see (20) with s = ft)
and Bjh_1 (see (23) with s = ft — 1). We note that the only difference is that J ; h>a in
the former is replaced by Jith-i m t n e latter. By (22), we see that Jjyh-i is the
elementary Pj-type interval of order ft —1 containing JJihiX. Hence, by (4), the
associated residue class R(m(j, ft, a), p1}) of JjKai is contained in the associated
ON IRREGULARITIES OF DISTRIBUTION
161
residue class, Ro say, of Jjth-i- So R o is the residue class modulo p)~l containing
R(m{j,h,a),phj).
Let Ju i = l,...,j-l,j
+ l,...,k,
denote any elementary
pt-type
interval of order h. Then we see that, since [0, ^ ^ - i ) and [0, £irh) are the union of a
finite number of mutually disjoint intervals of the type Jh it follows from (8) that to
prove (24), it suffices to prove that
£ £[«(t + aMhjphJ-1); JY x ... x j ^ x J J>M x JJ+1 x ... x Jk x /*]
a=0
= E[Q(t); Ji x ... x J j . i x Jj. h _! x JJ+1 x ... x Jk x /*] .
(26)
For i = 1,..., j— 1, j +1,..., /c, let Rt denote the associated residue class of J{. Then
the summand on the left-hand side of (26) is, by (1) and (7), equal to
1
= F\t+U,
n...nRj_1
nfi(m(j,/t,4p})nfi J . +
n ... n R7_, n (aM^-1
1
n ... n
+R(m(j, h, a),pj)) n Rj + 1 n ... n R J ; 7*1.
(27)
On the other hand,
X) (aM)phrl
a=0
+ R(m(j, h, a), p1})) = Ro .
V
(28)
7
It follows from (27) and (28) that the left-hand side of (26), and so also the righthand side, is equal to
F[t + (7?1 n . . . n R j . ! n Ro n RJ + 1 n ... n R t ) ; / * ] .
This completes the proof of (24). Equation (25) follows similarly, for if Rs and T ^ j
are respectively the associated residue classes of J, s and Jj s-i, then the analogue of
(28) is
This completes the proof of Lemma 4.
For s = l,...,h,
write, for the sake of simplicity,
G.(t)= X E[Q(t);BjJ.
(29)
To= [ EiaWiBt-Xdt;
(30)
Let
Ib2
W. W. L. CHEN
and, for w = 2 , . . . , W,
E[Q(t); BJ_ Jw-»G?(t)dt.
Tw=\
(31)
o
For j = 1,..., k, we write
Bj.Jdt;
(32)
further, for s = 1,..., h— 1 and w = 1,..., W — 1, we write
M*
0
Our proof of the Main Lemma is based on the following identity.
LEMMA 5.
We have, for h > 2, t/iat
w\
= To+ X [
)TW+W
fc-1 W-l
/W_1\
w
S,,
(34)
Proo/. We note, by (14), (15) and (16), that B* = B%; so, by taking W-th powers
on both sides of (19) with s = h, and using binomial expansion on the right-hand
side, we have, in view of (29),
E[_Q(t);B*r = ElD.(t);B*.,]w+W £ £[Q(t); flf-J^Ep^t); B,-J
By (30) and (31), we see that to prove Lemma 5, it remains to show that, for
h-l
~
V
J,hPj
v
^j + j,h L
W-\
Pj
L
ON IRREGULARITIES OF DISTRIBUTION
163
Now since B*_! is a box of class h—l with respect to the primes p l 7 . . . , pk, we know,
by Lemma 3, that E[Q(t); B%_ J is periodic in t with period Mh~l, and therefore also
periodic with period MjpJ" 1 , for every) = 1,..., k. So, by (21) with s = h and (24),
we have, for j = 1,..., k and writing Vj for the left-hand side of (35),
SI = 1
0
B,h_{\dt.
Now, by (19) and (29), we have, for s =
(36)
l,...,h-l,
£[n(t);B?_1]"'-H-1Gr(0.
(37)
On combining (36) and the case s = h — l of (37) and (33), we have, for j = 1,..., k,
V
J
=
For s = l,...,h — 2, we know that B* is a box of class s with respect to the primes
plt...,pk,
and so, by Lemma 3, (25), (37) and (33), we have, f o r ; = l,...,k and
s=
l,...,h-2,
=
Pjl\ E[Q{t)
o
l
^)Sj,,,v. (39)
In view of (32) the result (35) now follows from (38) by repeated application of (39).
This completes the proof of Lemma 5.
164
W. W. L. CHEN
§5. The case k = 1. The following lemma is only applicable for the case k = 1.
LEMMA 6.
For s = 1,..., h,
|G,(t)| = | £ [ n ( t ) ; B i , J < P i ;
(40)
|E[QU);B ljS ]|< 1.
(41)
also, for s = I, ...,h — 1,
Proof. Recall (21) with; = 1. For s = 1,..., h,
Since v l s < p 1 ; we see that to prove (40) it suffices to show that, for each
a = 1,..., v liS ,
|E[n(0;B1.,.J<l.
(42)
Now on recalling (20), we know, for s = l,...,h
and a = l , . . . , v l s , that
^i,s,c< = ^i, s ,a x ^*> where J1>s>ai is an elementary p1 -type interval of order s. It follows
from (1), (7) and the remark before (20) that
£[«(t);B!,,J =F[t + R(m(l,s,a),p\);I*'].
(43)
(42) now follows from (43) and (2). Inequality (41) follows similarly, since, for
s = l,...,h — l, we know, by (23) and (22), that Bls = Jlsxl*,
and Jls is an
elementary pj-type interval of order s. This completes the proof of Lemma 6.
We shall prove by induction on h that the Main Lemma for k = 1 holds for a
constant C satisfying
W) = (2w+lPl)w.
(44)
Note that, in particular,
C>p?.
(45)
Suppose B* = J t x / * , where / x = [0,1). Then by (1), (7) and (2),
and the Main Lemma for k = 1 is proved. We may therefore suppose that
Ix f [0,1), and hence, by (16), that B% = 0. For h = 1, we know, by (18) and (40),
that
|£[fi(t);B*]| =|£[Q(f); SJ]| = \E[Q{t);Bul]\ < Pl .
So in view of (45), the Main Lemma for k = 1 holds for h = 1. Suppose now that
h > 1 and that the Main Lemma for k = 1 holds, when h is replaced by any smaller
ON IRREGULARITIES OF DISTRIBUTION
positive integer, so that in particular, for any s = 2,...,h,
Lemma 3,
165
we have, in view of
B*_{]wdt ^ CM\s-\)iW.
(46)
Applying Holder's inequality, (40) and (46) to (31), we have, for w = 2,..., W,
in view of (45), so that
w
fW
TW < Cl-l'w2wp1Mh(h-l)^:1.
(47)
On the other hand, since B$ = 0, we have £ [ ^ ( t ) ; BJ] = 0, and so, by (32),
St = 0 .
(48)
Applying Holder's inequality, (40), (41) and (46) to (33), we have, for s = 2,..., h—l
and vv = 1,..., W — 1, in view of (45),
Furthermore
Si,i,w-i ^pY-'Mh
< C
For w = 1,..., W — 2, we have S l i l i W = 0. It follows that
W V . / E Pi"'*-'Y f ^ " 1 ) ^ , , , , < C'-'/^-1WPlM*(fc-l)*"'-1. (49)
On the other hand, by (30) and (46),
(50)
Combining (34), (47), (48), (49) and (50),
ElQ(t);B*-]wdt
=
CMh{(h-l)iw+
in view of (44). The proof of the Main Lemma for k = 1 is now complete.
166
W. W. L. CHEN
§6. The case k ^ 2. We suppose, for every j = 1,..., k, that the corresponding
Main Lemma for y-sets and boxes with respect to the primes p 1 ; ..., Pj-^, pJ + i, •••, pk
holds with Cj(p1, ...,pj_1 pJ+l,...,
pk, W) and Mj in place of C{pt,..., pk, W) and M
respectively, and with the exponent of h replaced by \{k — \)W. We let
C o = C0(p1, ...,pk, W) be defined by
C0 = l +
max Cj{p1,...,pj-1,pj+1,...,pk,
W).
(51)
1 <J j =S k
We shall prove, by induction on h, that the Main Lemma holds for a constant C
satisfying
C = C(Pl,...,Pk,
W) = 2kw2+w(Pl ...Pkr+1C0(Pl,...,Pk,
W).
(52)
Clearly
C>(Pi...pk)w+lC0
(53)
and
C>( P l ...p f c r-
(54)
First of all, we must establish a link which enables us to use the analogues of the
Main Lemma in lower-dimensional cases.
For j = \,...,k, s = \,...,h and a = l,...,v J S , we define the set Qj,s,a(t) of
points in fc-dimensional space by
% s ,*( f ) = {(*i(")> •••. Xj-i(n\ xj+l(n),...,
xk{n),n + t: n e R{m{j, s, a), p))} ,
where R(m(j, s, a), p*) is the associated residue class modulo p) of J, s a. Then, for
every real number t, we know, by Lemma 2, that QJS>a(t) is a translated pj-set of class
h with respect to the primes p l 5 ...,pJ_l,pJ+i,...,pk.
On the other hand, let Bfs be
defined for the above values of j and s by
B*s = [0, «!,,_!) x ... x [0, £,_!,,_!) x [0, ^ + 1 > s ) x ... x [0, £ til ) x /* .
Then BJ5 is a box of class s with respect to the primes pl,,.., Pj-lt Pj+1, •••, pk- It is
easy to see that, for j = 1,..., k, s = 1,..., h and a = 1,..., v JS ,
£["(0 ; BhsJ = E[ah,Jt); BfJ .
(55)
It follows, on our assumption of the lower-dimensional cases of the Main Lemma,
and hence its alternative form, in view of (55), (51), Lemma 3, (9) and (10), that for
every 7 = 1, --., k, s = 1,..., h and a = 1,..., v JS ,
pfMf
j
C0Mhsi{k-l)W.
(56)
ON IRREGULARITIES OF DISTRIBUTION
167
Similarly, for j= \,...,k
and s = l , . . . , / i - l , on relating E[Q(r);5 J S ] to the
discrepancy of certain translated Pj-set with respect to the primes p t , . . . , Pj_ l 5
pj+1,..., pk in the box Bfh, we have
E[a{t);BJjS]wdt
< C0Mhhiik-'l)W.
(57)
o
We know, by (29) and (21), that, for s =
l,...,h,
o
\W
k
k
vy,s M*
Vj s
,?! ' J ^ . ? , j £[fl(f); BUsJwdt;
(58)
o
so that, on combining (56) and (58), we have, for s = 1,..., h,
<( Pl ...p t r +I C 0 M* s *<*- 1 >«'.
(59)
We may assume that B* = i 1 x . . . x / t x P , where for some j = l,...,k,
Ij + [0,1).
Otherwise,
by
(1),
(7)
and
(2), we
know
that
|£[Q(£); B*]| = |F[t + Z ; / * ] | < 1, and the Main Lemma follows. By (16), we see
that B$ = 0. On the other hand, it is easy to see, from the definition of Q(t), that for
any fixed t and r\, E[Q(t); B*(i;, Y)] is periodic in Y with period px ... pk, so that we
may assume that 0 ^ Y < pl ... pk when estimating £[Q(t); BJ]. With this
restriction on Y, Z[Q.{t);Bf] ^ px ... pk and V(B^) < p1 ... pk, so, by (7),
|£[fi(f); B*]| ^ Pi ...pk- Hence, by (54), we see that the Main Lemma holds for
h = 1 with C defined by (52).
Suppose now that h ^ 2, and that the Main Lemma holds when h is replaced by
any smaller positive integer, so that, in particular, for s = 2,...,h, in view of
Lemma 3,
Bf.^dt
^ CM*(s-l)iW.
(60)
Applying Holder's inequality, (59) and (60) to (31), for w = 2,..., W, we obtain
Cl-[/wCl0/w(Pl
...
168
W. W. L. CHEN
by (53). Hence
w=2
(
(61)
W
On the other hand, since B% = 0 , we know that E[Q(t); Bg] = 0, and so by (32),
we deduce that for every j = 1,..., k,
Sj = O.
(62)
Applying Holder's inequality, (57), (59) and (60) to (33), for j =
s = 2 , . . . , h — 1 and w = 1,..., W — 1, in view of (53), we obtain
\,...,k,
Furthermore,
For j = l,...,/c and w = 1,..., W-2, we know that Sy l w = 0. It follows that
k
W-
h-1
^ Cl-1/wC10/w(Pl ...pkfw+"/w2kwMh(h-l)ikw~\
since 2*k'l)2w-lW
^
i{k >)+2W 2
2
-
-
(63)
< 2kw. Also, by (30) and (60), we deduce that
To ^ CMh(h-l)ikw
.
(64)
Combining (34), (61), (62), (63) and (64), in view of (52), we obtain
E[n(t);B*]wdt
o
= CMh{(h-l)*kW
+ {h-
This completes the proof of the Main Lemma.
§7. Proof of Theorem 2. We now take pt,...,
pk to be the first k primes. For any
ON IRREGULARITIES OF DISTRIBUTION
169
natural number N ^ 2, let
h = [log2 N] + 1 .
(65)
Then, for every j = 1,..., k, we have
Af < pj .
(66)
For any 0 = (61,..., 0k) in U\, and, for any real Y satisfying
0 < Y s£ N,
(67)
let B(0, Y) be the box defined by
B(0, Y) = [0, 0 J x ... x [0, 0*) x [0, 7 ) .
Let t] = »/(0) = (r]1,..., t]k) be defined such that, for every; satisfying 1 ^ j ^ k,
ij = ij(ej)=
-p;*[-pj0j,
(68)
i.e. r\j is the least integer multiple of pjh not less than 9j.
Let Q(f) = fi'(t), where H'(t) is defined by (3) in terms of the first Mh terms of the
Hammersley (Halton) sequence.
LEMMA 7.
For any 0 in U\ and any Y satisfying (67),
|£[Q(t); B{0, 7)] - £[Q(t); B*(V(0), Y)]| ^ k.
Proof. For j = 0,..., k, for any fixed 0 in (7* and for any fixed Y satisfying (67),
let
BU) = B O) ( 0 ; Y)
= [0, tiJx
... x [ 0 , r,j) x [ 0 , 6j+1)x...
x [ 0 , 0 , ) x [ 0 , Y).
Then clearly B(0) = B(6, Y) and B(k) = B*(ti(0), Y). It is easy to see that to prove
Lemma 7, it suffices to prove that, for every j = 1,..., k,
|£[Q(f);5 ( J ) ]-.E[n(r);B 0 - 1 ) ]| ^ 1.
For each j = 1,..., k, we know, by (68), that ^ > 9j and ij — Bj < pjh. It follows
from (66) and (67) that
0 «
By (7) with q = 1, it remains to show, for j = 1,..., k, that
0 «S Z[Q(t); B 0) ] - Z [ Q ( r ) ; B°~ x) ] ^ 1.
(69)
The first inequality of (69) is obvious. On the other hand, the difference in question
in (69) does not exceed the number of n in [ — t, Y — t) for which Xj(n) lies in
170
ON IRREGULARITIES OF DISTRIBUTION
[ij~Pj~h>rlj)- This interval is an elementary p,-type interval of order h, so that, in
view of (66) and (67), and, since Q'(£) is of class h, the difference is either 0 or 1. This
proves (69), and the proof of Lemma 7 is now complete.
For any even positive integer W, we know, by Lemma 7, that for any 0 in U\ and
any real Y satisfying 0 < Y ^ N,
E[Q(t);B(9,
Y)T ^ 2wE[Q(t); B*(ti(6), Y)~]w + (2k)w.
(70)
By the Main Lemma, we have
JV
I I (" £[Q(t); B*(>/(0), Y)']wdtd0l...d0kdY
0
k
V\
<KWMhh-kwN.
(71)
0
Combining (70) and (71), we see that there is a real number £*, satisfying
0 «S t* < M \ such that
, Y)~\wdBl...d9kdY
<KW {log N)*kwN ,
(72)
in view of (65). Note that the value of t* may depend on the value of W. On recalling
the definition of Q'(t) and (6), we see from (72) that the set
& = {(X1{n),...,Xk(n),N'1{n
+ t*)):0 < n + t* < N}
(73)
gives a proof of Theorem 2 for even positive integers W.
The case for general positive W follows immediately.
References
1. H. Davenport. "Note on irregularities of distribution", Mathematika, 3 (1956), 131-135.
2. J. H. Halton and S. K. Zaremba. "The extreme and I? discrepancies of some plane sets", Monatsh.fur
Math., 73 (1969), 316-328.
3. K. F. Roth. "On irregularities of distribution", Mathematika, 1 (1954), 73-79.
4. K. F. Roth. "On irregularities of distribution. II", Communications on Pure and Applied Math., 29
(1976), 749-754.
5. K. F. Roth. "On irregularities of distribution. Ill", Ada Aritk, 35 (1979), 373-384.
6. K. F. Roth. "On irregularities of distribution. IV", to appear in Ada Arith.
7. W. M. Schmidt. "Irregularities of distribution. X", Number theory and algebra, pp. 311-329 (Academic
Press, New York, 1977).
8. I. V. Vilenkin. "Plane nets of integration" (Russian), Z. Vycisl. Mat. i Mat. Fiz., 7 (1967), 189-196;
English translation in U.S.S.R. Comp. Math, and Math. Phys., 7(1) (1967), 258-267.
Dr. W. W. L. Chen,
Imperial College,
London SW7.
10F40: NUMBER
THEORY; Diophantine
proximation, Distribution modulo one.
Received on the 30th of May, 1980.
ap-