www.oeaw.ac.at
β-expansions in algebraic
function fields over finite
fields
K. Scheicher
RICAM-Report 2004-02
www.ricam.oeaw.ac.at
β-expansions in algebraic function fields over
finite fields
Klaus Scheicher
1
Introduction
In the present paper, we define a new kind of digit system in algebraic
function fields over finite fields. There are striking analogies of these digit
systems to the well known β-expansions defined in R+ . Results corresponding to classical theorems as well as open problems will be proved. In order
to pursue this analogy we will recall the definition of real β-expansions and
state the classical theorems corresponding to our results.
β-expansions of real numbers were introduced by Rényi [15]. Since then,
their arithmetic, diophantine and ergodic properties have been extensively
studied by several authors (cf. [1, 2, 7, 8, 9, 14, 17]). Given β > 1, the
β-transformation T = Tβ is defined for x ∈ [0, 1) by T (x) = βx − bβxc.
If β = b is an integer and if
x=
x1 x2 x3
+ 2 + 3 + . . . = .x1 x2 x3 . . . .
b
b
b
is the b-ary expansion of x, then T x = .x2 x3 . . .. Thus, T acts as a one-sided
shift on the sequence of b-ary digits of x.
If β is not an integer, we can use T to obtain an expansion for which
the above description is still valid. By iterating this map and considering
Date: April 7, 2004
2000 Mathematics Subject Classification: 11A63,11R06.
Key words and phrases: β-expansion, Pisot elements, greedy algorithm.
1
β-expansions in algebraic function fields
its trajectory
x
x
2
x
1
2
3
x −→
T (x) −→
T 2 (x) −→
...
with xj = bβT j−1 xc, we obtain
x=
x1 x2
x3
+ 2 + 3 + ....
β
β
β
We will call the sequence
dβ (x) = .x1 x2 . . .
the β-expansion of x. It is easy to prove that an infinite sequence of nonnegative integers (xi )i≥1 is the β-expansion of x ∈ [0, 1) if and only if
(1.1)
xi β −i + xi+1 β −i−1 + . . . < β −i+1
for every i ≥ 1. We say that dβ (x) is finite when xi = 0 for all sufficiently
large i. This is the case when there in an integer i ≥ 0 such that T i x = 0.
Now let x ≥ 1. Then there is an integer n such that β n ≤ x < β n+1 . We
define in a similar manner
dβ (x) = x−n . . . x−1 x0 .x1 x2 . . . .
Let
Per(β) = {x ∈ R+ : dβ (x) is eventually periodic} and
Fin(β) = {x ∈ R+ : dβ (x) is finite}.
Recall that a Pisot number is an algebraic integer β > 1 for which all
algebraic conjugates γ with γ 6= β satisfy |γ| < 1. A Salem number is an
algebraic integer β > 1 for which all algebraic conjugates γ with γ 6= β
satisfy |γ| ≤ 1 with at least one conjugate having |γ| = 1.
Theorem 1.1 (Bertrand and Schmidt [7, 17]) If β is a Pisot number,
then Per(β) = Q(β) ∩ R+ .
If β = b is an integer, then Q(b) = Q. Since every rational integer b > 1 is
a Pisot number, Theorem 1.1 is a natural generalization of the well known
fact that x ∈ Q if and only if db (x) is eventually periodic.
There exists a partial converse of Theorem 1.1.
β-expansions in algebraic function fields
3
Theorem 1.2 (Schmidt [17]) If Q ∩ R+ ⊂ Per(β), then β is a Pisot or
Salem number.
Remark 1.3 Schmidt stated his results for numbers in the unit interval.
It is trivial that Per(β) = Q(β) ∩ R+ ⇐⇒ Per(β) ∩ [0, 1) = Q(β) ∩ [0, 1)
and Q ∩ R+ ⊂ Per(β) ⇐⇒ Q ∩ [0, 1) ⊂ Per(β) ∩ [0, 1). We have changed
his notation in order to emphasize the analogies to our results.
Schmidt conjectured, that the converse of Theorem 1.2 is also true, i.e. if
β is a Salem number, then Q ∩ R+ ⊂ Per(β) (The Pisot case is trivially
included in Theorem 1.1). In the setting of algebraic function fields, we can
prove even more, since we can prove an analogue of Theorem 1.1 also for
Salem elements.
A similar situation occurs in the case of finite expansions. We say that
a number β > 1 has the finiteness property or property (F), if
Fin(β) = Z[β −1 ] ∩ R+ .
(F)
This property was introduced in [9].
Theorem 1.4 (Frougny and Solomyak [9])
(i) If Z+ ⊂ Fin(β), then β is a Pisot or Salem number.
(ii) Condition (F) implies that β is a Pisot number.
Several classes of Pisot numbers are known, such that (F) holds. On the
other hand, there exist examples of Pisot numbers, such that (F) is not
fulfilled (cf. [4, 9, 10]).
In the case of algebraic function fields, we are able to prove that no such
exceptional cases exist. Furthermore, we can prove that the analogues of
conditions in (i) and (ii) are equivalent.
This paper is organized as follows. In section two, we will define F((x−1 )),
the field of pole like formal Laurent series about ∞ as well as the analogues
to Pisot and Salem numbers in F((x−1 )). Furthermore, we will provide a
simple algorithm to compute the coefficients of Pisot and Salem elements in
F((x−1 )).
In section three, we will define the expansion algorithm for F((x−1 )) and
prove that there are no dependencies between consecutive digits.
Section four is devoted to periodic expansions. We will prove an extended
analogue of Theorem 1.1 as well as an analogue of Theorem 1.2.
β-expansions in algebraic function fields
4
In section five, we will give a complete characterization of all bases, which
give raise to finite expansions. In contrast, such a simple characterization
is hardly expected in the real case.
2
Pisot and Salem elements in a field of formal Laurent series over a finite field
Let F be a finite field, F[x] the ring of polynomials, F(x) the field of rational
functions. Let F((x−1 )) be the field of formal Laurent series of the form
(2.1)
z=
X̀
z k xk ,
zk ∈ F
k=−∞
where
½
` = deg z :=
Define the absolute value
|z| =
max{k : zk 6= 0} for
−∞ for
½
cdeg z for
0
for
z=
6 0
z = 0.
z=
6 0
z = 0.
where c > 1 is an arbitrary but fixed real number. Note that the set of
possible values of | · | is a discrete set. Then F((x−1 )) is the closure of F(x)
with respect to | · |.
Thus F[x] is the analogue to Z+ , F(x) is the analogue to Q and F((x−1 ))
is the analogue to R. If β ∈ F((x−1 )), then F(x, β) is the analogue to Q(β),
F[x, β] is the analogue to Z[β] and F[x, β −1 ] is the analogue to Z[β −1 ].
Since | · | is not archimedean, | · | fullfills the strict triangle inequality
|x + y| ≤ max(|x|, |y|) and |x + y| = max(|x|, |y|) if |x| 6= |y|. F((x−1 ))
together with | · | becomes a Banach space.
For a ∈ F((x−1 )) and r ∈ R+ , set
D(a, r) = {z ∈ F((x−1 )) : |z − a| < r}.
Let z be as in (2.1). Define the integer (polynomial) part
bzc =
X̀
k=0
zk xk
β-expansions in algebraic function fields
5
where the empty sum, as usual, is defined to be zero. Therefore bzc ∈ F[x]
and z − bzc ∈ D(0, 1) for all z ∈ F((x−1 )). Note that bz + wc = bzc + bwc,
b−zc = −bzc and |bzc| ≤ |z|. The following two definitions are due to [6].
Definition 2.1 An element β = β1 ∈ F((x−1 )) is called Pisot element if
it is an algebraic integer over F[x], |β| > 1 and |βj | < 1 for all Galois
conjugates βj .
Definition 2.2 An element β = β1 ∈ F((x−1 )) is called Salem element if it
is an algebraic integer over F[x], |β| > 1, |βj | ≤ 1 for all Galois conjugates
βj , and there exists at least one Galois conjugate βk , such that |βk | = 1.
In general, β and its Galois conjugates are hard to compute. Therefore, the
conditions in the Definitions 2.1 and 2.2 are difficult to verify. By considering
the Newton polygon (cf. [12, 13]) of the minimal polynomial, the following,
more useful equivalences [6, Theorem 12.1.1] can be derived.
Theorem 2.3 Let β ∈ F((x−1 )) be algebraic integer over F[x] and
(2.2)
p(y) = y n − a1 y n−1 − . . . − an ,
ai ∈ F[x]
be its minimal polynomial. Then
n
(i) β is a Pisot element if and only if deg a1 > 0 and deg a1 > max deg aj .
j=2
n
(ii) β is a Salem element if and only if deg a1 > 0 and deg a1 = max deg aj .
j=2
Remark 2.4 In [5, Theorem 1.1.] it is proved, that F((x−1 )) contains Pisot
elements of all degrees for arbitrary fields F.
In Theorem 2.6, a method to compute the coefficients of Pisot or Salem
elements is given. For the proof, we will need the following auxiliary result.
Lemma 2.5 If z, w ∈ F((x−1 )) with |z| = |w|, then |z n −wn | ≤ |z−w||z|n−1
for all n ∈ Z.
Proof. The statement is trivial for n = 0. For n > 0, we have
|z n − wn | = |z − w||z n−1 + z n−2 w + . . . + wn−1 |
n−1
≤ |z − w| max |z n−1−j wj |
j=0
n−1
= |z − w||z|
β-expansions in algebraic function fields
6
and
|z −n − w−n | = |wn − z n ||z −n w−n |
≤ |w − z||w|n−1 |z −n w−n |
= |z − w||z|−n−1 .
2
Theorem 2.6 Let β be a Pisot or Salem element and (2.2) be its minimal
polynomial. Then the recurrence
y1 = a1
yk+1 = a1 +
a2
an
+ . . . + n−1
yk
yk
for k ≥ 1
fulfills
lim yk = β.
k→∞
Proof. First we prove by induction that |yk | = |a1 | for all k ≥ 1. For k = 1
this assertion is trivial.
Let |yk | = |a1 | or equivalently, deg yk = deg a1 . For j = 2, . . . , n, it
follows from deg a1 > 0 and deg a1 ≥ deg aj that
deg aj /ykj−1 = deg aj − (j − 1)deg yk
≤ deg a1 − 1 deg a1
= 0 < deg a1 .
Thus deg yk+1 = deg a1 or equivalently |yk+1 | = |a1 |. From Lemma 2.5, we
get
¯ µ
¶
µ
¶¯
¯
¯
1
1
1
1
¯
¯
|yk+1 − yk | = ¯a2
−
+ . . . + an
n−1 − n−1 ¯
yk yk−1
yk
yk−1
µ
¶
|a2 |
|an |
≤ max
,...,
|yk − yk−1 |.
|a1 |2
|a1 |n
Since |a1 | > 1 and |a1 | ≥ |aj |, the left factor is constant and less than 1.
2
Thus, the sequence converges.
β-expansions in algebraic function fields
7
Example 2.7 Let p(y) = y 2 + xy + x over Z2 . Since deg a1 = deg a2 , its
zero must be a Salem element. Then the above sequence converges to
β = lim yk = x +
k→∞
∞
X
k=0
1
x2k −1
.
An elementary computation verifies that p(β) = 0. Since p(y) is quadratic,
Vieta’s formula implies
β2 = a1 − β =
∞
X
k=0
3
1
x2k −1
.
β-expansions in F((x−1))
Let β, z ∈ F((x−1 )) with |β| > 1, |z| < 1. A representation in base β (or
β-representation) of z is an infinite sequence (di )i≥1 , di ∈ F[x] such that
(3.1)
∞
X
di
z=
.
βi
i=1
A particular β-representation – called the β-expansion – can be computed
by a greedy algorithm.
This algorithm works as follows. Set r(0) = z and let dj = bβr(j−1) c,
r(j) = βr(j−1) − dj for j ≥ 1. This procedure yields a representation of z of
the form (3.1). Note that |dj | < |β| and |r(j) | < 1 for all j. The β-expansion
of z will be denoted by
dβ (z) = .d1 d2 . . . .
Note that
Ã
r(k) = β k
z−
k
X
!
dl β −l
.
l=1
An equivalent definition of the β-expansion is obtained by using the βtransformation on D(0, 1) which is given by the mapping
Tβ : D(0, 1) →
7
D(0, 1)
z →
7
βz − bβzc.
i−1
Then dβ (z) = (di )∞
i=1 if and only if di = bβTβ (z)c. Note that dβ (z) is finite
if and only if there is a k ≥ 0 such that Tβk (z) = 0.
β-expansions in algebraic function fields
8
Now let z ∈ F((x−1 )) be an element with |z| ≥ 1. Then there is a
unique k ∈ N such that |β|k ≤ |z| < |β|k+1 . Hence |z/β k+1 | < 1 and we
can represent z by shifting dβ (z/β k+1 ) by k digits to the left. Therefore, if
dβ (z) = .d1 d2 d3 . . ., then dβ (βz) := d1 .d2 d3 . . ..
In the sequel, we will use the following notations:
Per(β) = {z ∈ F((x−1 )) : dβ (z) is eventually periodic} and
Fin(β) = {z ∈ F((x−1 )) : dβ (z) is finite}.
It is clear that Fin(β) ⊂ Per(β) but Fin(β) 6= Per(β).
The following theorem provides an analogue to the condition mentioned
in (1.1).
Theorem 3.1 An infinite sequence of polynomials (dj )j≥1 is the β-expansion
of z ∈ D(0, 1) if and only if |dj | < |β| for j ≥ 1. Therefore, consecutive
digits of z are independent.
Proof. If (di )i≥1 is the β-expansion of z ∈ D(0, 1), then the expansion
algorithm implies that |di | < |β| for i ≥ 1.
Now let z ∈ D(0, 1) and z = d1 /β + d2 /β 2 + . . . be a β-representation
with |dj | < |β| for j ≥ 1. Let z = d01 /β + d02 /β 2 + . . . be the β-expansion of
z. Then
d1 − d01 d2 − d02
0=
+
+ ...
β
β2
with |di − d0i | < |β|. Suppose that not all summands are zero. Let i =
min{j : dj − d0j 6= 0}. Since
¯
¯
¯ di − d0i ¯
1
¯
¯
¯ β i ¯ ≥ |β|i
and
¯
¯
¯
µ¯
¶
¯ di+1 − d0i+1
¯
¯ di+1 − d0i+1 ¯
¯
¯,... < 1 ,
+ . . .¯¯ ≤ max ¯¯
¯
¯
i+1
i+1
β
β
|β|i
we run into a contradiction.
2
Remark 3.2 In contrast to the real case, there is no carry occurring, when
we add two digits. Therefore, if z, w ∈ F((x−1 )), we have dβ (z + w) =
dβ (z) + dβ (w) digitwise.
β-expansions in algebraic function fields
9
Example 3.3 Take p(y) from Example 2.7. Since β 2 + xβ + x = 0 and
1 = −1 in Z2 , we obtain
x=
β2
1
1
= β + 1 + + 2 + ....
β+1
β β
Thus dβ (x) = 11.11 . . .. Therefore, x ∈ Per(β) but x 6∈ Fin(β).
4
Periodic expansions
The aim of the current section is to prove Theorems 4.1 and 4.3. In the
case of Pisot elements, Theorem 4.1 provides an analogue to the classical
result of Bertrand and Schmidt [7, 17] mentioned in the introduction. The
corresponding problem on real β-expansions by Salem-numbers is still open.
Theorem 4.3 is an improved analogue of Schmidt’s partial converse of the
classical result.
Theorem 4.1 Let β be a Pisot or Salem element. Then
Per(β) = F(x, β).
Proof. The proof for Per(β) ⊂ F(x, β) is trivial. To prove F(x, β) ⊂ Per(β),
we mainly follow [11, Proposition 7.2.19].
It is sufficient to prove the result for F(x, β) ∩ D(0, 1). Let z ∈ F(x, β) ∩
D(0, 1). Then
n−1
X
−1
z=q
pi β i
i=0
with q, pi ∈ F[x] and deg q as small as possible. Let (dk )k≥1 be the βexpansion of z and
Ã
!
n−1
k
X
X
(k)
−`
k
−1
i
pi βj −
d` βj
(4.1)
rj = βj q
i=0
(k)
`=1
(k)
for j = 1, . . . , n. Therefore, r1 = r(k) and rj , j = 2, . . . , n are the Galois
conjugates of r(k) .
β-expansions in algebraic function fields
10
(k)
We have |r1 | = |r(k) | < 1 for all k. For j = 2, . . . , n, from |βj | ≤ 1 and
|d` | < |β| follows
³
´
k
(k)
k−`
k (0)
|rj | ≤ max |βj | |rj |, max(|d` βj |)
³
´ `=1
(0)
≤ max |rj |, |β| < ∞.
Here the strict triangle inequality has been applied (This is the crucial step
which doesn’t work in the setting of real β-expansions by Salem-numbers).
(k)
Therefore, |rj | is bounded for all k and j. We need a technical result.
(k)
(k)
Lemma 4.2 Let R(k) = (r1 , . . . , rn ) and B = (βj−i )1≤i,j≤n . Then for
(k)
(k)
every k ≥ 0, there exists a unique n-tuple W (k) = (w1 , . . . , wn ) ∈ F[x]n
such that R(k) = q −1 W (k) B.
Proof. We use induction by k. Note that β n = a1 β n−1 + . . . + an . Thus
r
(0)
= z = q
−1
n−1
X
pi β i
i=0
Ã
!
(0)
(0)
w
w
n
1
= q −1
+ ... + n .
β
β
Now r(k+1) = βr(k) − dk+1 , and hence
Ã
!
(k)
(k)
w
wn
(k)
r(k+1) = q −1 w1 + 2 + . . . + n−1 − qdk+1
β
β
Ã
!
(k+1)
(k+1)
w
w
n
1
= q −1
+ ... +
.
β
βn
(k)
Since q, wi
of r(k) .
(k)
∈ F[x], the induction also works for the Galois conjugates rj
2
Now we proceed with the proof of Theorem 4.1. Let H (k) = qR(k) . Since
(k)
|rj | is bounded for every j, the sequence kH (k) k is bounded. As the matrix
B is invertible, for every k ≥ 1,
(k)
(k)
kH (k) B −1 k = kW (k) k = k(w1 , . . . , wn(k) )k = max |wj | < ∞.
1≤j≤n
β-expansions in algebraic function fields
11
Thus there exist p and m such that W (m+p) = W (m) , and therefore, r(m+p) =
r(m) which implies that the β-expansion of z is eventually periodic.
2
The following Theorem is the converse of Theorem 4.1.
Theorem 4.3 Let F[x] ⊂ Per(β). Then β is a Pisot or Salem element.
Remark 4.4 In Theorem 1.2, Q ∩ R+ ⊂ Per(β) is needed. The analogue
to this condition would be F(x) ⊂ Per(β) which is, of course, stronger than
F[x] ⊂ Per(β). Thus, Theorem 4.3 is an improved analogue of Theorem 1.2.
Proof. From
bβc
β − bβc
=1−
β
β
and |β − bβc| < 1 we obtain
bβc = β +
d1 d2
+ 2 + ...
β
β
where |di | < |β|.
Since bβc ∈ F[x], the expansion of bβc must be eventually periodic. Therefore,
bβc = β +
d1
dk
dk+1
dk+p
dk+1
dk+p
+ . . . + k + k+1 + . . . + k+p + k+p+1 + . . . + k+2p + . . . .
β
β
β
β
β
β
Thus
µ
β
k
d1
dk
bβc − β −
− ... − k
β
β
¶
µ
=β
k+p
d1
dk+p
bβc − β −
− . . . − k+p
β
β
¶
.
If the β-expansion of bβc is finite, the right hand side of this equation is
zero. In both cases β must be an algebraic integer.
Suppose that β has a Galois conjugate βj 6= β with |βj | > 1. Choose m
with
µ ¯ ¯¶
¯β¯
m
m
|β − βj | > max 1, ¯¯ ¯¯ .
βj
Since
bβ m c
β m − bβ m c
=
1
−
βm
βm
β-expansions in algebraic function fields
12
and |β m − bβ m c| < 1, we have
bβ m c = β m +
d1 d2
+ 2 + ...
β
β
where |di | < 1.
Since bβ m c ∈ F[x], the expansion must be eventually periodic. Consider the
(k)
β-expansion of z = bβ m c − β m ∈ D(0, 1). Let r(k) , rj be as in the proof of
Theorem 4.1. Equation (4.1) yields
k
X
d` r(k)
bβ c = β +
+ k
β`
β
`=1
m
m
for k ≥ 0.
Since bβ m c ∈ F[x] and βj is an Galois conjugate of β, it must fulfill the
equation
(k)
k
X
d` rj
m
m
bβ c = βj +
+ k for k ≥ 0.
`
β
βj
j
`=1
Since the expansion of z is eventually periodic, the r(k) take only finitely
(k)
many values. Thus, the same is true for rj . Hence, limk→∞ r(k) /β k = 0. If
(k)
|βj | > 1, then limk→∞ rj /βjk = 0. Therefore,
Ã
!
∞
X
1
1
d`
−
β m − βjm +
= 0.
β ` βj`
`=1
From
¯∞
¯ ¯!
Ã
Ã
!¯
¯ ¯
¯X
¯ ¯
1 ¯¯
1
∞ ¯ d` ¯
∞ ¯ d` ¯
¯
¯ ¯ , max ¯ ¯
max
d`
−
≤
max
¯
¯
`=1 ¯ β ` ¯ `=1 ¯ βj` ¯
¯
β ` βj` ¯
`=1
µ ¯ ¯¶
¯β¯
< max 1, ¯¯ ¯¯
βj
we get a contradiction.
2
We can combine Theorem 4.1 and Theorem 4.3 to obtain
Corollary 4.5 An element β ∈ F((x−1 )) is a Pisot or Salem element if and
only if
F[x] ⊂ Per(β).
β-expansions in algebraic function fields
5
13
Finite expansions
The present section is devoted to the characterization of Fin(β), the set of
finite expansions. Contrary to the case of real β-expansions, we can prove a
complete characterization result in our setting. We will need the following
Lemma 5.1 Let β be an arbitrary element of F((x−1 )) with deg β > 0, and
let z ∈ F[x, β −1 ] have a purely periodic β-expansion. Then z ∈ F[x, β].
Proof. Assume z ∈ F[x, β −1 ] is purely periodic with period n. Let dβ (z) =
.d1 d2 . . .. Since z ∈ F[x, β −1 ], there is a m such that β mn z ∈ F[x, β]. Therefore
z = β mn z − d1 β mn−1 − . . . − dmn ∈ F[x, β].
2
Theorem 5.2 Let β ∈ F((x−1 )) be a Pisot element. Then
Fin(β) = F[x, β −1 ].
(F)
Remark 5.3 Note that (F) is true if and only if for every z ∈ F[x, β −1 ],
there is a k ≥ 0 such that T k (z) = 0.
Proof. Since it is trivial that Fin(β) ⊂ F[x, β −1 ], we need to prove only the
other direction. Let
β n − a1 β n−1 − . . . − an = 0
(5.1)
with deg a1 > deg aj for j > 1.
From Theorem 4.1 it follows that F[x, β −1 ] ⊂ F(x, β) ⊂ Per(β). Thus
z ∈ F[x, β −1 ] has an eventually periodic expansion. Therefore, it can be
decomposed into z = zf + zp , where dβ (zf ) is finite and dβ (zp ) is purely
periodic. Hence, by Lemma 5.1, zp ∈ F[x, β]. Since
r
(k)
k
= β (zf + zp ) −
k
X
dl βjk−l ,
l=1
there is an integer k such that r(k) ∈ F[x, β], and we can restrict our attention
to F[x, β]. For i = 1, . . . , n, let
(5.2)
(5.3)
vi := β i−1 − a1 β i−2 − . . . − ai−1
ai
an
=
+ . . . + n−i+1
β
β
β-expansions in algebraic function fields
14
and V = {v1 , . . . , vn }. Note that v1 = 1.
Then V is a base of F[x, β] over F[x]. Hence, for every z ∈ F[x, β], there
are z1 , . . . , zn ∈ F[x] such that
z = z 1 v1 + . . . + z n vn .
For
z = z̃0 + . . . + z̃n−1 β n−1 ∈ F[x, β],
let
ẽ(z) = [z̃0 , . . . , z̃n−1 ]Tβ
the vector of coordinates with respect to the standard base {1, β, . . . , β n−1 }.
Analogously, we will denote the vector of coordinates with respect to V by
e(z) = [z1 , . . . , zn ]TV .
Using (5.2), the coordinates with respect to V can be computed from the
standard coordinates from the system





z̃0
1 −a1 · · · −an−1
z1
..
...
 .. 

  .. 
.
 . 
 0 1
 . 
 .  = . .

 .
. . . . . −a1   ... 
 .. 
 ..
z̃n−1 β
0 ···
0
1
zn V
If we define ak = 0 for k > n and using (5.3), we can write
à n
!
n
X
X
(5.4)
z=
aj zj+i−1 β −i .
i=1
j=1
In base V , multiplication by β is represented by

a1 a2 · · · an−1
 1 0 ···
0

..
...

.
M = 0 1
 . .
.
..
..
 ..
0
0 ··· 0
1
the matrix

an
0 
.. 

. .

0 
0
Define the vectors v = [v1 , . . . , vn ]T and e = [1, 0, . . . , ]T . We consider the
greedy algorithm for z ∈ F[x, β] with respect to V . Write z = e(z). Since
Tβ (z) = βz − bβzc,
β-expansions in algebraic function fields
15
we can write the β-transformation in base V as
T : F[x, β] →
7
F[x, β]
z →
7
M z − (bM z · vc)e.
Furthermore we have e(Tβ (z))) = T (e(z)), which shows that the following
diagram is commutative:
F[x, β]
Tβ
/ F[x, β]
e
e
²
F[x]n
²
T
/ F[x]n .
With respect to V , we can express T as follows:



z1
−bz1 v2 + . . . + zn−1 vn c
 z2 

z1



T :  ..  → 
..
 . 

.
zn V
zn−1



 .

V
Thus F[x, β] together with T provides an analogon to the so called shift
radix system defined in [3]. However, due to our notation, the indices here
are in the reverse direction as in [3].
For j = 2, . . . , n it follows that
deg
aj
β
= deg aj − deg β
< deg a1 − deg a1 = 0
and therefore
deg vj < 0.
Now
(5.5)
n−1
deg (−bz1 v2 + . . . + zn−1 vn c) ≤ max deg (zi vi+1 )
i=1
n−1
< max deg zi .
i=1
For k ≥ 0, let
(k)
z (k) = (z1 , . . . , zn(k) )TV = e(T k (z)).
β-expansions in algebraic function fields
16
(k)
(k)
Thus, (F) is true if and only if there is a k ≥ 0, such that (z1 , . . . , zn )T =
(0, . . . , 0)T or equivalently, since z (k) ∈ F[x]n ,
n
(k)
max deg zi
i=1
= 0.
It follows from (5.5) that
³
´
n−1
(1)
(0)
deg z1 ≤ max deg zi − 1 .
i=1
From
(1)
(1)
(0)
(0)
(z1 , . . . , zn(1) )T = (z1 , z1 , . . . , zn−1 )T ,
we obtain
(2)
deg z1
³
´
n−1
(1)
≤ max deg zi − 1
i=1
³ n−1
´
n−2
(0)
(0)
= max max deg zi − 2, max deg zi − 1
i=1
³ i=1
´
(0)
(0)
(0)
= max deg z1 − 1, . . . , deg zn−2 − 1, deg zn−1 − 2 .
Analogously
(3)
deg z1
³
´
n−1
(2)
≤ max deg zi − 1
i=1
³
´
(0)
(0)
(0)
(0)
= max deg z1 − 1, . . . , deg zn−3 − 1, deg zn−2 − 2, deg zn−1 − 2 .
After n − 2 such steps we obtain
´
³
(0)
(0)
(0)
(n−1)
deg z1
≤ max deg z1 − 1, deg z2 − 2, . . . , deg zn−1 − 2 .
Thus
n
(n)
max deg zi
i=1
n
(n+1−i)
= max deg z1
i=1
³
´
n
(0)
≤ max deg zi − 1 .
i=1
(h)
Going on this way, we will find a number h such that maxni=1 deg zi
and we are done.
=0
2
The following theorem forms the converse of Theorem 5.2.
β-expansions in algebraic function fields
17
Theorem 5.4 If F[x, β −1 ] ⊂ Fin(β), then β is a Pisot element.
Proof. Applying arguments from complex analysis, the proof of the corresponding result for real β-expansions [9, Lemma 1 (b)] is pretty short.
Unfortunately, this technique doesn’t work in our context. Therefore, we
will adapt the idea of [16, Lemma 2.4].
Suppose that
n
max deg aj ≥ deg a1 .
j=2
We will construct an element z ∈ F[x, β] which does not have a finite representation. Let vi be as in (5.2). Define
n
i0 = i0 (a1 , . . . , an ) := max{ i : deg ai = max deg aj }
j=1
and
(
j0 =
j0 ((z1 , . . . , zn )TV )
:=
n
n
min{ i : deg zi = max deg zj } if max deg zj > 0,
j=1
j=1
∞
otherwise.
Select
(5.6)
(0)
z = z (0) := (z1 , . . . , zn(0) )TV
with j0 (z (0) ) + 1 ≤ i0
(take for instance z (0) := (x, 0, . . . , 0)TV ). Thus
1 < i0 ≤ n and 1 ≤ j0 < n.
Furthermore,
deg vi0 = deg ai0 − deg β ≥ 0,
(0)
n−1
(0)
deg zj0 = max deg zj > 0,
j=1
deg vi < deg vi0 for i > i0 ,
(0)
(0)
deg zj < deg zj0 for j < j0 .
We will show that z (0) has an infinite representation by proving that
(5.7)
j0 (z (k) ) + 1 ≤ i0
for all k ≥ 0.
By the definition of j0 , this implies that z (k) 6= 0 for all k ≥ 0 and we are
done. Let
(k)
z (k) = (z1 , . . . , zn(k) )TV .
β-expansions in algebraic function fields
18
We will prove (5.7) by induction.
Since (5.7) holds for k = 0 by (5.6), we can proceed to the induction
step.
Suppose that (5.7) holds for a certain k and note that
(5.8)
(k+1)
z (k+1) = (z1
(k+1)
, . . . , zn(k+1) )TV = (z1
(k)
(k)
, z1 , . . . , zn−1 )TV .
We distinguish two cases.
Case 1. j0 := j0 (z (k) ) < i0 − 1. By (5.8) and because j0 < n, we have
n
(k+1)
max deg zj
j=1
(k)
(k+1)
= max(deg zj0 , deg z1
(k+1)
)
(k+1)
= max(deg zj0 +1 , deg z1
) > 0.
Thus j0 (z (k+1) ) = 1 or j0 (z (k+1) ) = j0 + 1. Both of these inequalities imply
that
j0 (z (k+1) ) ≤ i0 − 1
and we are done.
Case 2. j0 := j0 (z (k) ) = i0 − 1. The definitions of i0 and j0 imply that
deg vi0 ≥ 0,
deg vi < deg vi0 for i > i0 ,
(k)
deg zi0 −1 > 0,
Thus
(k)
deg zj
(k)
(k)
< deg zi0 −1 for j < i0 − 1.
(k)
deg (zi0 −1 vi0 ) > deg (zi−1 vi ) for i 6= i0 .
This implies that no cancellations occur in the highest power of x in the
sum
(k)
(k)
z1 v2 + . . . + zn−1 vn .
Hence,
(k)
(k)
(k)
deg (z1 v2 + . . . + zn−1 vn ) = deg zi0 −1 + deg vi0
> 0,
and therefore
(k)
(k)
(k)
(k)
deg (−bz1 v2 + . . . + zn−1 vn c) = deg (z1 v2 + . . . + zn−1 vn )
(k)
= deg (zi0 −1 vi0 )
(k)
≥ deg zi0 −1
n−1
(k)
= max deg zj .
j=1
β-expansions in algebraic function fields
This implies that
(k+1)
deg z1
n
(k+1)
≥ max deg zj
j=2
19
.
Thus,
j0 (z (k+1) ) = 1 ≤ i0 − 1
and we are done also in this case.
2
It turns out that condition (F) is equivalent to a seemingly weaker condition.
Theorem 5.5 F[x, β −1 ] ⊂ Fin(β) if and only if F[x] ⊂ Fin(β).
Proof. Of course, if F[x, β −1 ] ⊂ Fin(β), then F[x] ⊂ Fin(β). To prove the
converse, consider an element
z1
z`
+ . . . + ` ∈ F[x, β −1 ] where zi ∈ F[x].
β
β
P
P
There exist finite expansions zi = j dij /β j . Therefore zi /β i = j dij /β i+j .
Adding up the corresponding digits, we obtain the β-expansion of z, which
is again finite.
2
We can combine Theorem 5.2 together with Theorem 5.4 and Theorem 5.5 to obtain
z = z0 +
Corollary 5.6 An element β ∈ F((x−1 )) is a Pisot element if and only if
F[x] ⊂ Fin(β).
Remark 5.7 The real analogue of condition F[x] ⊂ Fin(β) is Z+ ⊂ Fin(β).
In the case of real expansions, one can only prove that β is a Pisot or Salem
number. Thus Corollary 5.6 is an improved analogue of Theorem 1.4 (i).
References
[1] S. Akiyama. Pisot numbers and greedy algorithm. In Number theory
(Eger, 1996), pages 9–21. de Gruyter, Berlin, 1998.
[2] S. Akiyama. Self affine tiling and Pisot numeration system. In Number
theory and its applications (Kyoto, 1997), volume 2 of Dev. Math.,
pages 7–17. Kluwer Acad. Publ., Dordrecht, 1999.
β-expansions in algebraic function fields
20
[3] S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, and J. M. Thuswaldner.
Generalized radix representations and dynamical systems I. submitted.
[4] Shigeki Akiyama. Cubic Pisot units with finite beta expansions. In
Algebraic number theory and Diophantine analysis (Graz, 1998), pages
11–26. de Gruyter, Berlin, 2000.
[5] P. T. Bateman and A. L. Duquette. The analogue of the PisotVijayaraghavan numbers in fields of formal power series. Ill. J. Math.,
6:594–606, 1962.
[6] M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. PathiauxDelefosse, and J. P. Schreiber. Pisot and Salem numbers. Birkhäuser
Verlag, Basel, 1992.
[7] A. Bertrand. Développements en base de Pisot et répartition modulo
1. C. R. Acad. Sci. Paris Sér. A-B, 285(6):A419–A421, 1977.
[8] F. Blanchard. β-expansions and symbolic dynamics. Theoret. Comput.
Sci., 65(2):131–141, 1989.
[9] C. Frougny and B. Solomyak. Finite beta-expansions. Ergodic Theory
Dynam. Systems, 12(4):713–723, 1992.
[10] M. Hollander. Linear numeration systems, finite beta expansions, and
discrete spectrum of substitution dynamical systems. PhD thesis, University of Washington, 1996.
[11] M. Lothaire. Algebraic combinatorics on words, volume 90 of Encyclopedia of Mathematics and its Applications. Cambridge University
Press, Cambridge, 2002.
[12] J. Neukirch. Algebraic number theory, volume 322 of Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
[13] H. Niederreiter and C. Xing. Rational points on curves over finite
fields: theory and applications, volume 285 of London Mathematical
Society Lecture Note Series. Cambridge University Press, Cambridge,
2001.
β-expansions in algebraic function fields
21
[14] W. Parry. On the β-expansions of real numbers. Acta Math. Acad. Sci.
Hungar., 11:401–416, 1960.
[15] A. Rényi. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar, 8:477–493, 1957.
[16] K. Scheicher and J. M. Thuswaldner. Digit systems in polynomial rings
over finite fields. Finite Fields Appl., 9(3):322–333, 2003.
[17] K. Schmidt. On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc., 12(4):269–278, 1980.
Klaus Scheicher,
Johann Radon Institute for
Computational and Applied Mathematics,
Austrian Academy of Sciences,
Altenbergerstraße 69,
A-4040 Linz, Austria
[email protected]