Prime numbers, Determinism and Pseudorandomness L -estimation of - - PowerPoint PPT Presentation

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Prime numbers, Determinism and Pseudorandomness L∞-estimation of Generalized Thue-Morse Trigonometric Polynomials and Dynamical maximization

Aihua Fan (Amiens/Wuhan) J¨

• rg Schmeling (Lund)

and Weixiao Shen (Shanghai) CIRM, 5 November 2019

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Outline

1

Problems

2

Known facts

3

Ergodic maximization

4

Theory of q-Sturmian measures

5

Sturmian conditions

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Problems

Problems

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Problems

Digital sum function

q ≥ 2: integer. q-expansion of n ∈ N: n = ∑

finite

ϵjqj. (0 ≤ ϵj < q). Digital sum: Sq(n) = ∑

finite

ϵj. q-additivity: ∀0 ≤ n < qk Sq(mqk + n) = Sq(mqk) + Sq(n).

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Problems

Generalized Thue-Morse sequences

Definition t(q,c)

n

:= t(c)

n

:= e2πic·Sq(n), (q ≥ 2; c ∈ [0, 1)). Thue-Morse sequence (q = 2, c = 1/2): (−1)S2(n) +/ − / − +/ − + + −/ − + + − + − − +/ · · · (Key property) q-multiplicative function f : N → C: ∀n < qt, f (mqt + n) = f (mqt)f (n).

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Problems

Generalized Thue-Morse Trigonometric Polynomials

(Generalized) Thue-Morse trigonometric polynomials σ(q,c)

N

(x) := σ(c)

N (x) := N−1

∑

n=0

t(c)

n e2πinx.

Gelfond (1968): For q = 2, c = 1/2, ∥σ(2,1/2)

N

∥∞ = O(N

log 3 log 4 ),

where log 3/ log 4 is best possible (very nice proof!). Problem: How to estimate the norms ∥σ(c)

N ∥p (1 ≤ p ≤ ∞) ?

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Problems

Gelfond integers: Motivation I

Let m, p, z, a, b be integers with m ≥ 2, p ≥ 2, z ≥ 2. Define T0(x) = #{n ≤ x : Sq(n) = a mod p, n = b mod m}; T1(x) = #{n ≤ x : Sq(n) = a mod p, pz ̸ |n} . Theorem (Gelfond 1968) If (p, q − 1) = 1, there exists 0 < λ < 1 independent from m, a, b such that T0(x) = x mp + O(xλ). λ =? If p is prime, there exists 0 < λ1 < 1 such that T1(x) = x pζ(z) + O(xλ1). λ1 =?

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Problems

Good weights-Davenport exponent:Motivation II

The Davenport exponent of (wn), denoted H((wn)), is the best h > 0 such that sup

t∈[0,1)

• N−1

∑

n=0

wne2πint

• = Oh(N/ logh N).

Theorem (Fan 2017) If (wn) ∈ ℓ∞ with H((wn)) > 1

2, then it is L1-good for a.e. convergence.

Actually, for any MPDS (X, B, ν, T) and and f ∈ L1(ν), we have ν-a.e. lim

N→∞

1 N

N−1

∑

n=0

wnf (T nx) = 0.

1

Sarnak: M¨

• bius weight; El Abdalaoui et al: H > 1.

2

Proof based on Davenport-Erd¨

• s-LeVeque (1965): limN 1

N

∑N

n=1 ξn = 0 a.s.

if ∥ξn∥∞ = O(1), ∑ ∥ξ1+···+ξn∥2

2

n3

< ∞.

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Problems

Very good weights-Gelfond exponent: Motivation II

The Gelfond exponent of (wn), denoted D((wn)), is the best d > 0 such that sup

n0≥0

sup

t∈[0,1)

• n0+N−1

∑

n=n0

wne2πint

• = Od(Nd).

Theorem (Fan 2017) Suppose (wn) ∈ ℓ∞ and D := D((wn)) < 1. Let (X, B, ν, T) be a MPDS and f ∈ L2(ν). Then ν-a.e.

N−1

∑

n=0

wnf (T nx) = o(ND(log N)2(log log N)1+ϵ).

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Problems

Problems

Problems to which we get answers:

1 Find the best exponent γ(q, c) := γ(c) for which

∥σ(c)

N ∥∞ = O(Nγ(c)).

2 (Multifractally ) analyze the size of the sets

{x ∈ R : |σ(c)

qn (x)| ∼ qnα}, α ∈ R.

Observation. ∥σ(c)

N ∥∞ ≥ ∥σ(c) N ∥2 =

√ N, so γ(c) ≥ 1/2. It is also not difficult to show γ(c) < 1. γ(1/2) = log 3

log 4, the only known case (Gelfond).

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Known facts

Known facts

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Known facts

About γ(q, c), c ̸= 0

1 2 ≤ γ(q, c)< 1.

Mauduit-Rivat-Sarkozy (2017): γ(2, c) ≤ π2∥c∥ 20 log 2. Fan-Koniezny (2018): Gowers uniform norms Assume f : N → {z ∈ C, |z| = 1} be q-multiplicative.Then ∀s ≥ 2, ∃τs > 0, ∥f ∥Us ≪ ∥f ∥τs

U1.

So, γ(q, c) < 1 implies: ∀P ∈ R[x], ∃γ′ < 1. ∥ ∑N−1 t(q,c)

n

e2πiP(n)x∥∞ = O(Nγ′).

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Known facts

Subsequences of TM sequence

Theorem (Mauduit-Rivat, 2010/2009) For any α ∈ (0, 1), there exists σ(α) > 0 such that ∑

p≤N,p∈P

e2πiαS2(p) ≪ N1−σ(α). For any α ∈ (0, 1), there exists c(α) > 0 such that ∑

n≤N

e2πiαS2(n2) ≪ N1−c(α)(1 + log log N)5.

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Ergodic maximization

Ergodic maximization

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Ergodic maximization

Dynamical interpretation

σ(c)

2m (x) = 1

∑

a0=0 1

∑

a1=0

· · ·

1

∑

am−1=0

e2πic∑m−1

j=0 aje2πi∑m−1 j=0 aj2jx.

=

m−1

∏

j=0

( 1 + e2πi(c+2jx)) |σ(c)

2m (x)| = 2m m−1

∏

j=0

| cos(π(2jx + c))|. Let fc(x) = log | cos(π(x + c))| and let T(x) = 2x mod 1. Then 1 m log |σ(c)

2m−1(x)| = log 2 + 1

m

m−1

∑

j=0

fc(T j(x)).

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Ergodic maximization

For q ≥ 2, we have Tx = qx mod 1 and fc(x) = log

• sin πq(x + c)

q sin π(x + c)

• .

Figure: The graph of f0 on the interval [−1/q, 1 − 1/q], here q = 6.

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Ergodic maximization

Ergodic maximization

Observation: fc is upper semi-continuous. M: the set of all T-invariant Borel probability measures on T = R/Z. We have β(c) = sup {∫ fcdµ : µ ∈ M } . By Birkhorff Ergodic Theorem, γ(c) ≥ 1 + β(c) log q .

• Fact. We actually have equality. (Conze-Guivarc’hunpublished, Jenkinson)

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Ergodic maximization

Ergodic maximization-general setting

Let T : M → M be a (topological) dynamical system. Let f : M → R be a measurable function. Ergodic optimization studies the quantity sup {∫

M

fdµ : µ ∈ MT } . A probablity measure is called a maximizing measure for (T, f ) if it attains the supremum. Ma˜ n´ e, Conze-Guivarc’h, Bousch, Jenkinson, Contreras et al: mostly in the case that T is the doubling map on T and f is Lipschitz. Problem Maximizing mesaure (Existence? Uniquenee? Kind?)

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Ergodic maximization

Some known results

Let T(x) = 2x mod 1 be the doubling map on T = R/Z. Ma˜ n´ e: Suppose f is Lipschitz. Then

1

∃ a maximizing measure µmax and it is revealed: ∃u : T → R Lipschitz, such that for ψ := f − (u − u ◦ T), supp µmax ⊂ {x : ψ(x) = ∥ψ∥∞}.

2

ψ(x)−β = maxy:2y=x(f (y)+ ψ(y)) admits a solution (ψ, β) ∈ Lip ×R.

Bousch: For f (x) = cos 2π(x + θ), there is a unique maximizing measure which is supported on a closed half circle. (Sturmian measure). Jenkinson: If f is strictly concave, then there is a unique maximizing

• measure. (Sturmian measure).

A Sturmian measure is an invariant mesure supported by a semi-circle.

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Ergodic maximization

Our main result: Maximizing measure for fc

Theorem (FSS) For each c, fc has a unique maximizing measure µc. µc is a Sturmian measure supported in some [λ, λ + q−1] (∃λ ∈ [−c − q−1, −c]). For c outside a set of Hausdorff dimension zero, µc is periodic. β(c) can be (numerically) computed for many c’s.

• Examples. q = 2

µ1/2 is supported on the 2-cycle {1/3, 2/3}; β(c) = 1

2 log | cos 2π(c + 1/3) cos 2π(c + 2/3)|, c near 1/2.

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Ergodic maximization

Graph of γ(c)

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Theory of q-Sturmian measures

Theory of q-Sturmian measures

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Theory of q-Sturmian measures

q-Sturmian measures: Existence and Uniqueness

Notation: q ≥ 2, Tx = qx mod 1, Cγ = [γ, γ + q−1] mod 1. Theorem For each γ ∈ R, there is a unique T-invariant Borel probability measure Sγ supported in Cγ. We have Sγ = Sγ+1 (∀γ ∈ R). Moreover, there is a continuous, monotone map ρ : R → R such that (1) ρ(γ + 1) = ρ(γ) + q − 1. (defining a degree q − 1 map on T) (2) ρ(γ) ∈ Q if and only if Sγ is supported on a periodic orbit. (3) for each r ∈ Q, ρ−1(r) is a non-degenerate interval. (4) dim Sγ = 0 for all γ. (5) {γ ∈ [0, 1) : ρ(γ) ̸∈ Q} has upper Minskowski dimension zero. Definition: Sγ is the (q-)Sturmian measure associated to Cλ.

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Theory of q-Sturmian measures

Rotation number ρ(γ), Branch T|Cγ, Map Rγ

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Theory of q-Sturmian measures

When ρ(γ) is irrational

Let γ =

∞

∑

j=1

aj qj ∈ [0, 1), aj ∈ {0, 1, . . . , q − 1}. Then (1) a1 < q − 1 and for each j ≥ 2, aj = a1 or aj = a1 + 1; (2) for any n ≥ 1, [n(ρ(γ) − a1)] = #{1 ≤ j ≤ n : aj = a1 + 1} =

n

∑

j=1

ηj where ηj = 0, 1 according to aj = a1 or a1 + 1.

• N. B. γ → ρ(γ) → (ηn) → γ

It follows dim{γ ∈ [0, 1) : ρ(γ) ̸∈ Q} = 0.

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Sturmian conditions

Proof: Sturmian conditions

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Sturmian conditions

Pre-Sturmian condition and Sturmian condition

Notation: Tx = qx, Cγ = [γ, γ + q−1], τγ : [0, 1] → Cγ (inverse of T|Cγ). Definition Let f : T → [−∞, +∞) Lipschtzian on Cγ. f satisfies the pre-q-Sturmian condition for γ, if ∃ψ ∈ Lip(T, R), ∃β ∈ R ∀x ∈ Cγ, f (x) − ( ψ ◦ T(x) − ψ(x) ) = β. (1) f satisfies the q-Sturmian condition for γ, if furthermore, ∀y ∈ T \ Cγ, f (y) − ( ψ ◦ T(x) − ψ(x) ) < β, (2)

Facts (1) (Candidate ψ) Under Pre-Sturm condition, β = ∫ fdSγ and a.e. ψ′(x) =

∞

∑

n=1

f ′

c (τ n γx)

qn . (3) (2) Under Sturm condition, Sγ is the unique maximization measure.

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Sturmian conditions

fc: Pre-Sturmian condition/Sturmian condition

First leaving time of Cγ eγ(x) := inf{k ≥ 0 : T kx ∈ T \ Cγ} =

∞

∑

n=0

χτ n

γ(Cγ)(x).

Theorem (Bousch) Let f : T → [−∞, +∞) be Lipschtzian on Cλ. pre−Sturm condition for γ ⇐ ⇒ vf (γ) := ∫

Cλ

f ′(x)eγ(x)dx = 0. Theorem (FSS) (1) fc satisfies pre-Sturm condition for some γ. (2) actually fc satisfies Sturm condition.

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Sturmian conditions

Checking of Pre-Sturm condition/Sturm condition

∃γ ∈ (−c − q−1, −c) such that ∫

Cγ f ′ c(x)eγ(x)dx = 0 ?

R ∋ γ → eγ(·) ∈ L1(T) is continuous, even almost Lipschitz.

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Sturmian conditions

First leaving function eγ(x) (q = 2, γ = 1

4):

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Sturmian conditions November 5, 2019 31 / 32

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Sturmian conditions

Thank you for your attention!

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