Automatic sequences, generalised polynomials & nilmanifolds - - PowerPoint PPT Presentation

Automatic sequences, generalised polynomials & nilmanifolds

Jakub Konieczny

Univeristy of Oxford Jagiellonian University Hebrew University of Jerusalem

8 Kongres Polskiego Towarzystwa Matematycznego Lublin, 22 September 2017

Motivation

Theorem (Kronecker) Let θ ∈ Rd. Then, the sequence {nθ} = ({nθ1} , . . . , {nθd}) for n ∈ N is equidistributed in [0, 1)d unless there exists k ∈ Zd \ {0} such that k · θ ∈ Z. Notation: {x} = x − ⌊x⌋ for x ∈ R; {x} = ({x1} , . . . , {xd}) for x = (x1, . . . , xd) ∈ Rd; x · y = x1y1 + · · · + xdyd for x, y ∈ Rd. A sequence xn ∈ X (n ∈ N) is equidistributed with respect to µ ∈ Meas(X) iff for any f ∈ C(X): 1 N

• n<N

f(xn) →

• fdµ

as N → ∞. If X = [0, 1)d, µ is the Lebesgue measure by default. Remark If k ∈ Zd \ {0} and k · θ ∈ Z then k · {nθ} ∈ Z for all n ∈ Z.

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Motivation

Theorem (Weyl) Let p(x) ∈ R[x]d. Then, the sequence {p(n)} = ({p1(n)} , . . . , {pd(n)}) for n ∈ N is equidistributed in [0, 1)d unless there exists k ∈ Zd \ {0} such that k · p(x) ∈ Z[x]. Remark Identify [0, 1)d with the d-torus Rd/Zd. For k ∈ Zd \ {0}, the set

• x ∈ Rd/Zd : k · x ∈ Z
• is a (d − 1)-torus.

A more precise statement is true: If p(x) ∈ R[x]d then {p(n)} is equidistributed in a union of subtorii of Rd/Zd. What about more general expressions? Example √ 2 √ 3n

• is equidistributed in [0, 1).

√ 2 √ 2n

• is not equidistributed in [0, 1); and nor is

√ 2n √ 2n

• .

√ 2 √ 2n 2 is equidistributed in [0, 1).

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Motivation

Example √ 2 √ 3n

• is equidistributed in [0, 1).

Proof: √ 2 √ 3n

• =

√ 6n

• −

√ 2 √ 3n

• and

√ 6n

• ,

√ 3n

• is

eqdistributed in [0, 1)2. Example √ 2 √ 2n

• and

√ 2n √ 2n

• are not equidistributed in [0, 1).

Proof: √ 2 √ 2n

• =
• −

√ 2 √ 2n

• ;
• 2

√ 2n √ 2n

• =

√ 2n 2 and √ 2n

• is eqdistributed in [0, 1).

Example √ 2 √ 2n 2 is equidistributed in [0, 1). Proof: √ 2 √ 2n 2 =

• −
• 2

√ 2n2 + √ 2 √ 2n 2 and

• 2

√ 2n2 , √ 2n

• is eqdistributed in [0, 1)2.

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Generalised polynomials

Definition The generalised polynomial maps Z → R (denoted GP) are the smallest family such that R[x] ⊂ GP and for any g, h ∈ GP: g + h ∈ GP, g · h ∈ GP, ⌊g⌋ ∈ GP. (Here: ⌊g⌋ (n) = ⌊g(n)⌋.) If g ∈ GP then also {g} ∈ GP. (Here, {g} (n) = {g(n)}.) The generalised polynomial maps Z → Rd are just d-tuples (g1, . . . , gd) with gi ∈ GP for 1 ≤ i ≤ d. We call a set E ⊂ N a generalised polynomial set if 1E ∈ GP. Example

1 2 {n/2}

√ 2n2 is equidistributed with respect to 1

2δ0 + 1 2λ[0,1).

2 ⌊(n + 1)θ⌋ − ⌊nθ⌋ is equidistributed with respect to (1 − θ)δ0 + θδ1.

(0 < θ < 1)

3 √

2n 2 is equidistributed with respect to measure µ with dµ =

dt 2 √ t.

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Generalised polynomials

Question

1 For which g ∈ GP is {g} equidistributed in [0, 1)? 2 What are possible distributions of bounded g ∈ GP? 3 What may a generalised polynomial set look like?

Answers:

1 For generic choice of g ∈ GP, {g} is equidistributed in [0, 1). There is

an algorithmic way to decide (→ Haland-Knutson, Leibman).

2 Distribution of any generalised polynomial is described by an algebraic

expression, except for a few exceptional points (→ Bergelson, Leibman).

3 A generalised polynomial set with positive density is always

combinatorially rich (→ Bergelson, Leibman). Conversely, a sparse generalised polynomial set is always combinatorially poor. Any sufficiently sparse set is generalised polynomial (→ Byszewski, K.).

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Equidistributed generalised polynomials

The multiset of coefficients of a generalised polynomial g consists of the constants used to write down g(x). For instance, the coefficients of g0(n) = √ 2 √ 3n 2 + √ 5n √ 7n

• are

√ 2, √ 3, √ 5, √ 7. More precisely: If g(n) = αni then coeff(g) = {α}; coeff(g + h) = coeff(g) ∪ coeff(h); coeff(g ⌊h⌋i) = coeff(g) ∪ coeff(h). Theorem (Haland–Knutson) Let g ∈ GP be a generalised polynomial with coefficients α1, . . . , αr. Suppose that all products

i∈I αi for I ⊂ {1, 2, . . . , r} are linearly independent over

• Q. Then {g(n)} is equidistributed in [0, 1).

For instance, {g0(n)} is equidistributed because 1, √ 2, √ 3, √ 5, √ 7, √ 6, √ 10, √ 14, √ 15, √ 21, √ 35, √ 30, √ 42, √ 70, √ 105, √ 210 are Q-linearly independent.

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Nilmanifolds, nilsequences, and Nil–Bohr sets

Definition (nilsequences) Let G be a (d-step) nilpotent Lie group, and Γ < G a cocompact discrete subgroup.

1 The space X = G/Γ is a nilmanifold. 2 For g ∈ G, the map Tg : X → X, x → gx is a nilrotation. 3 The dynamical system (G/Γ, Tg) is a nilsystem. It has a natural Haar

measure µG/Γ which is Tg-invariant.

4 If F : X → R is a (smooth) function, x0 ∈ X, then ψ(n) = F(gnx0) is a

(d-step) nilsequence. A reassuring example: Take G = R, Γ = Z. Then G/Γ = T, the unit circle, equipped with rotations x → x + θ. The additive characters n → e(nθ) are 1-step nilsequences. Definition (Nil–Bohr sets) Let ψ be a (d-step) nilsequence, and V ⊂ R an open set. A set A = {n : ψ(n) ∈ V } is called a (d-step) Nil–Bohr set (if = ∅). If ψ(0) ∈ V (i.e. 0 ∈ A), then A is called a Nil–Bohr0 set.

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Nilmanifolds and generalised polynomials

Theorem (Bergelson–Leibman) Let g : Z → Rd be a bounded generalised polynomial. Then, there exists a minimal nilsystem (G/Γ, T) with a point x0 ∈ G/Γ and a piecewise polynomial map F : G/Γ → Rd such that g(n) = F(T nx0). Slogan: bounded GP sequences ≃ nilsequences; GP sets ≃ Nil–Bohr sets. Definition: A semialgebraic set V ⊂ [0, 1)m is one which can be defined by a finite number of polynomial equalities and inequalities. A map P : [0, 1)m → R is piecewise polynomial if there exists a partition [0, 1)m =

i Vi into semialgebraic pieces, such that P|Vi is a polynomial for

each i. A nilmanifold G/Γ carries a natural system of coordinates (→ Malcev basis), under which it can be identified with [0, 1)m for some m. Hence, it makes sense to speak of piecewise polynomial map G/Γ → Rd.

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Nilmanifolds and generalised polynomials

Theorem (Bergelson–Leibman) Let g : Z → Rd be a bounded generalised polynomial. Then, there exists a minimal nilsystem (G/Γ, T) with a point x0 ∈ G/Γ and a piecewise polynomial map F : G/Γ → Rd such that g(n) = F(T nx0). Definition: If A ⊂ Rm is semialgebraic with non-empty interior and P : A → Rd is polynomial, then we call S = P (A) a parametrized polynomial set. In this situation S carries a natural measure µS, the pushforward of the normalised Lebesgue measure on A. A parametrised piecewise polynomial set S ⊂ Rd is a finite union of polynomial sets S =

i Si, equipped with measure µS = i αiµSi, αi > 0.

Corollary Let g : Z → Rd be a bounded generalised polynomial. Then, there exists a parametrised piecewise polynomial set S ⊂ Rd with measure µS, such that g(n) ∈ S for all most all n and g(n) is equidistributed in S with respect to µS.

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IP sets

Finite sums. For n = (ni)∞

i=1, ni ∈ N, define:

FS(n) =

i∈α

ni : α ⊂ N, finite, α = ∅

• .

A set A ⊂ N is IP is there is n with A ⊃ FS(n). A set B ⊂ N is IP∗ if B ∩ A = ∅ for any IP set A. Fact

1 Any IP∗ set is syndetic (i.e. intersects any sufficiently long interval). 2 If (X, T) is a distal dynamical system, x ∈ U ⊂ X with U open, then

the set {n ∈ N : T nx ∈ U} is IP∗. Theorem (Hindman) If A is an IP set, A = A1 ∪ A2 ∪ · · · ∪ Ar then ∃j : Aj is IP. If B1, B2 . . . , Br are IP∗ sets then B = B1 ∩ B2 ∩ · · · ∩ Br is IP∗.

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IP sets and GP sets

Corollary (Bergelson–Leibman) Let g : Z → Rd be a bounded generalised polynomial. Then for almost all n ∈ N, for any δ > 0, the set {m ∈ N : |g(n + m) − g(n)| < δ} is IP∗. In particular, if A ⊂ N is a GP set then A − n is IP∗ for almost all n ∈ A. Proof. Represent g(n) = F(T nx0), as in Bergelson–Leibman Theorem. Because F is piecewise polynomial, F is continuous almost everywhere. In particular, for almost every n, F is continuous at T nx0. Restrict to such n. Any nilsystem is distal. Hence, for any open U ∋ T nx0, the set

• m ∈ N : T m+nx0 ∈ U
• is IP∗. Pick U so that |F(x) − F(y)| < δ for

all x, y ∈ U.

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Sparse GP sets

Question: Can we say anything about sparse GP sets? Definition: For A ⊂ N, d(A) = limN→∞ |A ∩ [0, N)| /N. We say that A is sparse iff d(A) = 0. Lemma Suppose that g(n) is a generalised polynomial and a < b. Then, the sets A = {n ∈ N : g(n) = 0} and B = {n ∈ N : a < g(n) < b} are generalised polynomial sets. Proof: To prove the first part note that for almost all λ ∈ R, {λg(n)} = 0 if and only if g(n) = 0. Hence, 1A(n) = 1 − ⌈{λg(n)}⌉. The second part is analogous. Example Let Fi denote the i-th Fibonacci number. Then {Fi : i ∈ N} is a generalised polynomial set. Proof: Using basic theory of continued fractions, notice that n is among the Fibonacci numbers if and only if nϕ < 1/2n, where x = min {|x − a| : a ∈ Z}. Apply the Lemma to g(n) = nnϕ.

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Recursive sequences

Example Let Ti be the i-th Tribonacci number, defined by T0 = T1 = 0, T2 = 1 and Ti+3 = Ti+2 + Ti+1 + Ti. Then {Ti : i ∈ N} is a generalised polynomial set. Proof strategy: Use theory of best Diophantine approximations in dimension 2 for (suitably chosen) Pisot numbers of degree 3. There exists β ∈ R, a Pisot number of degree 3, and a norm N on R, such that δn := min

• N
• n(β−1, β−2) − (a, b)
• : (a, b) ∈ Z2

takes minimal values exactly for n = Ti. More precisely, δTi ∼ β−i/2 and δn > δTi if n < Ti. Use this to encode {Ti : i ∈ N} by generalised polynomial conditions. Question Does there exist a degree ≥ 4 recursive sequence ai such that {Si : i ∈ N} is a generalised polynomial set? Is there any λ ∈ R, algebraic of degree ≥ 4 such that the set

• λi + 1/2
• : i ∈ N
• is a generalised polynomial set?

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Extremely sparse sequences

Example Suppose that ai is a sequence with lim infi→∞

log ai+1 log ai

> 1. Then, {ai : i ∈ N} is a generalised polynomial set. Proof strategy: Aim to find α ∈ R and C so that n = ai for some i if and

• nly if

αn < 1/nC (†) It is not hard to show that there exist (a Cantor set of) α such that (†) holds for all ai. The hard part is to show (†) holds for no more n. We first reduce to the case when D < log ai+1/ log ai < 2D for a large constant i. We strengthen (†) to 1/2nC < αn < 1/nC (‡) (the easy part works just like before). Under suitable conditions on C and D, we use continued fractions to check that no spurious n satisfy (‡).

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Sparse GP and combinatorics

Theorem (Byszewski, K.) Let A ⊂ N be a GP set with d(A) = 0. Then, there exists no n ∈ Z such that A − n contains an IP set. Proof outline:

1 Suppose the claim is false. Use Bergelson–Leibman Theorem to find a

nilsystem (G/Γ, Tg), a point x ∈ G/Γ and a piecewise polynomial set S ⊂ G/Γ such that S has empty interior but the set {n : gnx ∈ S} is IP.

2 Use S to construct an algebraic set R ⊂ G with empty interior which is

preserved under multiplication by {gn} for n in an IP set.

3 Show that there exists no proper subgroup H of G which contains {gn}

for infinitely many n. Derive a contradiction.

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Automatic sequences

Setup: k ∈ N, Σk = {0, 1, . . . , k − 1}, Σ∗

k = words over Σk.

An finite automaton A base k consists of the following data:

1 A finite set of states S; 2 A distinguished initial state s• ∈ S; 3 A transition function δ : S × Σk → S; 4 An output function τ : S → Ω for some finite set Ω.

Extend δ to : S × Σk → S by δ(s, uv) = δ(δ(s, u), v); denote base k expansion of n ∈ N by (n)k. The finite automaton A produces an k-automatic sequence given by aA(n) = τ (δ (s•, (n)k)) Example Thue-Morse sequence is the 2-automatic sequence given by t(n) = (−1)s2(n) where s2(n) denotes the sum of binary digits.

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Automatic sequences

Definition: Let Σ be a finite set. Any map ϕ: Σ → Σ∗ can be extended to a morphism Σ∗ → Σ∗ or Σ∞ → Σ∞ (denoted again ϕ) by the rule: w0w1w2 . . . → ϕ(w0)ϕ(w1)ϕ(w2) . . . A pure morphic word w ∈ Σ∗ is a fixed point of a morphism ϕ: ϕ(w) = w. If |ϕ(v)| = k for all v ∈ Σ, then ϕ is k-uniform. A coding is simply a map τ : Σ∞ → Ω∞ acting coordinate-wise: w0w1w2 . . . → τ(w0)τ(w1)τ(w2) . . . Fact A sequence a = a(0)a(1) . . . is k-automatic iff it is the coding of a fixed point of a k-uniform morphism.

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Automatic sequences

General problem: Given a class of sequences, decide which among them are k-automatic. 1Primes and 1Squares are not k-automatic for any k; Cobham Theorem: If log k/ log l ∈ Q, then the only l-automatic which are also k-automatic are eventually periodic; Allouche-Shallit: If α ∈ R \ Q, β ∈ R and m ∈ N≥2 then ⌊αn + β⌋ mod m is not k-automatic. Question Which among the generalised polynomial sequences are k-automatic? Example The Fibonacci word w ∈ {0, 1}∞ can be defined in the following ways:

1 w0 = 0 and w is fixed by the morphism ϕ: {0, 1}∞ → {0, 1}∞ given by

1 → 0 and 0 → 01;

2 wn = ⌊−(n + 2)ϕ⌋ + ⌊−(n + 1)ϕ⌋ + 2; 3 if n =

i≥2 aiFi where Fi is the i-th Fibonacci number

(F0 = 1, F1 = 1), ai ∈ {0, 1}, aiai+1 = 0, then wn = a2.

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Automatic sequences vs. generalised polynomials

Theorem (Byszewski, K.) Let p(x) ∈ R[x] and m ∈ R≥2. Then ⌊p(n)⌋ mod m is not k-automatic. Theorem (Byszewski, K.) Let g be a generalised polynomial. If g is also k-automatic, then there exists a set Z ⊂ N and a periodic sequence b such that g(n) = b(n) for n ∈ N \ Z and Z is small in the sense that supM∈N |Z ∩ [M, M + N)| = O(logC N) for a constant C. In fact, Z is arid. Proof ideas:

Fist show weaker statement: d(Z) = 0. Express g dynamically as g(n) = F(T n

g x0) in nilsystem (G/Γ, Tg). Assume that Tg is totally minimal.

Because g is k-automatic, there are t, r, r′ (r = r′) s. t. g(ktn + r) = g(ktn + r′). Using total minimality, conclude that F = const a.e. Reduce to the case when g(n) ∈ {0, 1}, and g(n) = 0 for almost all n. Show that a k-automatic set either contains IP set (actually, IPS), or is very sparse (arid). Use the previous theorem: sparse GP sets contain no IP sets.

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Automatic sequences vs. generalised polynomials

Theorem (Byszewski, K.) Fix k. Then one of the following holds:

1 All k-automatic are generalised polynomials are eventually periodic; 2 The set

• ki : i ∈ N
• f powers of k is generalised polynomial.

Proof idea: Bootstrap the previous theorem. Question: Which one is it?

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The End

Thank You for your attention!

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