Periodic representations in Salem bases
Tomáš Vávra
Numeration 2018

May 22, 2018

Periodic representations in Salem bases

Numeration 2018

(β, A)-representation is an expression in the form+∞Xai β −i ,ai ∈ Ai=−k

A representation is called eventually periodic if (ai )i∈N is eventuallyperiodic

Periodic representations in Salem bases

Numeration 2018

(β, A)-representation is an expression in the form+∞Xai β −i ,ai ∈ Ai=−k

A representation is called eventually periodic if (ai )i∈N is eventuallyperiodicTheorem (V.)
Let β, |β| > 1, be an algebraic number, then there exists A ⊂ Z
such that every element of Q(β) admits an eventually periodic(β, A)-representation.

Periodic representations in Salem bases

Numeration 2018

Theorem (K. Schmidt, 1980)
Let Per(β) denote the set of numbers with eventually periodic(greedy) β-expansions. Thenβ Pisot=⇒

Per(β) = Q(β) ∩ R+Per(β) = Q(β) ∩ R+=⇒

β Pisot or Salem

Periodic representations in Salem bases

Numeration 2018

Theorem (K. Schmidt, 1980)
Let Per(β) denote the set of numbers with eventually periodic(greedy) β-expansions. Thenβ Pisot=⇒Per(β) = Q(β) ∩ R+Per(β) = Q(β) ∩ R+=⇒β Pisot or Salem

Schmidt conjectured: Per(β) = Q(β) holds for Salem numbers
although a little is known (and that little is thanks to D. Boyd)

Periodic representations in Salem bases

Numeration 2018

Theorem (Baker, Kala, Masáková, Pelantová, V., 2017)
Let β be an algebraic number without conjugates on the unit
circle. Then Q(β) admits eventually periodic representations withsome (integer) alphabet.

Periodic representations in Salem bases

Numeration 2018

Theorem (Baker, Kala, Masáková, Pelantová, V., 2017)
Let β be an algebraic number without conjugates on the unit
circle. Then Q(β) admits eventually periodic representations withsome (integer) alphabet.

Theorem (Baker, Masáková, Pelantová, V., 2017)
If each x ∈ Q(β) ∩ Br (0) for some r > 0 has an eventually periodicP
representation of the form x = i≥1 ai β −i , then |β 0 | = |β| or|β 0 | ≤ 1 holds for all the conjugates β 0 of β.

Periodic representations in Salem bases

Numeration 2018

P. Savický, J. Šíma: Quasi-Periodic β-expansions and Cut
Languages, Theoret. comp. sci., 2018
P. Savický: Eventually periodic representations for β ≈ 1.722084
Salem root of x 4 − x 3 − x 2 − x + 1 and A = {−2, −1, 0, 1, 2}

Periodic representations in Salem bases

Numeration 2018

Representations in our case are made by the following procedure(Thurston 1989):
I let V ⊂ C be bounded
I let A ⊂ C be finite
I let T (x) = βx − D(x) : V → V ,(meaning that βV ⊂SD(x) ∈ Aa∈A (V + a))then every x ∈ V has a (β, A)-representation

Periodic representations in Salem bases

P+∞

i=1 ai β−iNumeration 2018

Representations in our case are made by the following procedure(Thurston 1989):
I let V ⊂ C be bounded
I let A ⊂ C be finite
I let T (x) = βx − D(x) : V → V ,(meaning that βV ⊂SD(x) ∈ Aa∈A (V + a))then every x ∈ V has a (β, A)-representationP+∞

i=1 ai β−iGreedy expansions: V = [0, 1), D(x) = bβxc

Periodic representations in Salem bases

Numeration 2018

If x ∈
/ V , then find n ∈ N such that β −n x ∈ V

Periodic representations in Salem bases

Numeration 2018

If x ∈
/ V , then find n ∈ N such that β −n x ∈ V
If T n (x) = T m (x) for some m, n ∈ N, then we get an eventuallyperiodic representation

Periodic representations in Salem bases

Numeration 2018

Pisot caseLet β be Pisot, β 0 its conjugate, i.e. |β 0 | < 1.
Take any x ∈ Q(β) ∩ V and start iterating T (x)We have:
I T (x) = βx − a ∈ V in the identical embedding
I |(T (x))0 | = |β 0 x 0 − a0 | < C eventually in other embeddings

Periodic representations in Salem bases

Numeration 2018

Pisot caseLet β be Pisot, β 0 its conjugate, i.e. |β 0 | < 1.
Take any x ∈ Q(β) ∩ V and start iterating T (x)We have:
I T (x) = βx − a ∈ V in the identical embedding
I |(T (x))0 | = |β 0 x 0 − a0 | < C eventually in other embeddings

Hence (T n (x))n∈N is bounded in each embedding

Periodic representations in Salem bases

Numeration 2018

Pisot caseLet β be Pisot, β 0 its conjugate, i.e. |β 0 | < 1.
Take any x ∈ Q(β) ∩ V and start iterating T (x)We have:
I T (x) = βx − a ∈ V in the identical embedding
I |(T (x))0 | = |β 0 x 0 − a0 | < C eventually in other embeddings

Hence (T n (x))n∈N is bounded in each embedding(T n (x))n∈N takes values in a lattice =⇒ T n (x) = T m (x)eventually
Periodic representations in Salem bases

Numeration 2018

Non-Pisot caseTwo problems:
I If |β 0 | ≥ 1, then (T n (x))0 might be unbounded
I If β is a noninteger, then T n (x) does not take values in alattice

Periodic representations in Salem bases

Numeration 2018

Non-Pisot caseTwo problems:
I If |β 0 | ≥ 1, then (T n (x))0 might be unbounded
I If β is a noninteger, then T n (x) does not take values in a

lattice
Solution: We define V bounded in embeddings where β is big, i.e.
I Embeddings corresponding to |β 0 | ≥ 1
I p-adic embeddings where |β|p > 1

Periodic representations in Salem bases

Numeration 2018

Yet another problem

If |β 0 | = 1, then we cannot do β −n x ∈ V

Periodic representations in Salem bases

Numeration 2018

Yet another problem

If |β 0 | = 1, then we cannot do β −n x ∈ V
But ifβ 0 B1 (0) ⊂[(B1 (0) + a),a∈Athenβ 0 Bm (0) ⊂[(Bm (0) + a)a∈Aholds for any m > 1 (with the same alphabet!)

Periodic representations in Salem bases

Numeration 2018

General caseYβ −n x ∈ Vm :=p|∞|β|p >1B1 (0) ×YBm (0) ×YOp ⊂ Kβp-∞
|β|p >1p|∞|β|p =1

For each individual set Vp in the product we can find Ap such thatSβVp ⊂ a∈Ap (Vp + a)
Then with A =Q

p Ap we have βVm ⊆Periodic representations in Salem bases

Sa∈A (Vm + a)

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We can find an approximation of A by Q(β) while the covering
still works

Periodic representations in Salem bases

Numeration 2018

We can find an approximation of A by Q(β) while the covering
still works
Moreover, for (complex) Salem bases, all x ∈ Q(β) ∩ B1 (0) have aP−i (weak-greedy property)
representation of the form +∞i=1 ai β

Periodic representations in Salem bases

Numeration 2018

We can find an approximation of A by Q(β) while the covering
still works
Moreover, for (complex) Salem bases, all x ∈ Q(β) ∩ B1 (0) have aP−i (weak-greedy property)
representation of the form +∞i=1 ai βWe cannot get weak-greedy property for non Pisot and non Salem
bases (and their complex analogy): start with x big in someembedding where |β|p > 1

Periodic representations in Salem bases

Numeration 2018

Question #2

Given β and A, can we tell whether Q(β) admits eventuallyperiodic (β, A)-representations?

Periodic representations in Salem bases

Numeration 2018

Theorem (V.)
Let β have no conjugate on the unit circle. The following
statements are equivalent
1. Q(β) admits eventually periodic (β, A)-representations
2. Z[β] admits eventually periodic (β, A)-representations
3. The set
X A (β) = {a−k β k + · · · + a−1 β + a0 : ai ∈ A, k ∈ N}
is relatively dense in Kβ
4. 0 ∈ int(K ) ⊂ Kβ , where βK =Periodic representations in Salem bases

Sa∈A (K + a)
Numeration 2018