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

Prime numbers, Determinism and Pseudorandomness L ∞ -estimation of Generalized Thue-Morse Trigonometric Polynomials and Dynamical maximization Aihua Fan (Amiens/Wuhan) J¨ org Schmeling (Lund) and Weixiao Shen (Shanghai) CIRM, 5 November 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 1 / 32

Outline Problems 1 Known facts 2 Ergodic maximization 3 Theory of q -Sturmian measures 4 Sturmian conditions 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 2 / 32

Problems Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 3 / 32

Problems Digital sum function q ≥ 2: integer. q -expansion of n ∈ N : ∑ ϵ j q j . n = finite (0 ≤ ϵ j < q ). Digital sum: ∑ S q ( n ) = ϵ j . finite q -additivity: ∀ 0 ≤ n < q k S q ( mq k + n ) = S q ( mq k ) + S q ( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 4 / 32

Problems Generalized Thue-Morse sequences Definition t ( q , c ) := t ( c ) := e 2 π ic · S q ( n ) , ( q ≥ 2; c ∈ [0 , 1)) . n n Thue-Morse sequence ( q = 2 , c = 1 / 2): ( − 1) S 2 ( n ) + / − / − + / − + + − / − + + − + − − + / · · · (Key property) q -multiplicative function f : N → C : f ( mq t + n ) = f ( mq t ) f ( n ) . ∀ n < q t , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 5 / 32

Problems Generalized Thue-Morse Trigonometric Polynomials (Generalized) Thue-Morse trigonometric polynomials N − 1 σ ( q , c ) ( x ) := σ ( c ) t ( c ) ∑ n e 2 π inx . N ( x ) := N n =0 Gelfond (1968): For q = 2 , c = 1 / 2, log 3 ∥ σ (2 , 1 / 2) log 4 ) , ∥ ∞ = O ( N N where log 3 / log 4 is best possible ( very nice proof! ). Problem : How to estimate the norms ∥ σ ( c ) N ∥ p (1 ≤ p ≤ ∞ ) ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 6 / 32

Problems Gelfond integers: Motivation I Let m , p , z , a , b be integers with m ≥ 2 , p ≥ 2 , z ≥ 2. Define T 0 ( x ) = # { n ≤ x : S q ( n ) = a mod p , n = b mod m } ; p z ̸ | n } T 1 ( x ) = # { n ≤ x : S q ( n ) = a mod p , . Theorem (Gelfond 1968) If ( p , q − 1) = 1 , there exists 0 < λ < 1 independent from m , a , b such that T 0 ( x ) = x mp + O ( x λ ) . λ =? If p is prime, there exists 0 < λ 1 < 1 such that x p ζ ( z ) + O ( x λ 1 ) . T 1 ( x ) = λ 1 =? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 7 / 32

Problems Good weights-Davenport exponent:Motivation II The Davenport exponent of ( w n ), denoted H (( w n )), is the best h > 0 such that � N − 1 � � � = O h ( N / log h N ) . � ∑ w n e 2 π int sup � � � � t ∈ [0 , 1) � n =0 Theorem (Fan 2017) If ( w n ) ∈ ℓ ∞ with H (( w n )) > 1 2 , then it is L 1 -good for a.e. convergence. Actually, for any MPDS ( X , B , ν, T ) and and f ∈ L 1 ( ν ), we have ν -a.e. N − 1 1 ∑ w n f ( T n x ) = 0 . lim N N →∞ n =0 Sarnak: M¨ obius weight; El Abdalaoui et al: H > 1. 1 ∑ N os-LeVeque (1965): lim N 1 Proof based on Davenport-Erd¨ n =1 ξ n = 0 a.s. 2 N ∑ ∥ ξ 1 + ··· + ξ n ∥ 2 if ∥ ξ n ∥ ∞ = O (1) , < ∞ . 2 n 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 8 / 32

Problems Very good weights-Gelfond exponent: Motivation II The Gelfond exponent of ( w n ), denoted D (( w n )), is the best d > 0 such that � n 0 + N − 1 � � � ∑ w n e 2 π int � = O d ( N d ) . sup sup � � � � n 0 ≥ 0 t ∈ [0 , 1) � n = n 0 Theorem (Fan 2017) Suppose ( w n ) ∈ ℓ ∞ and D := D (( w n )) < 1. Let ( X , B , ν, T ) be a MPDS and f ∈ L 2 ( ν ). Then ν -a.e. N − 1 ∑ w n f ( T n x ) = o ( N D (log N ) 2 (log log N ) 1+ ϵ ) . n =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 9 / 32

Problems Problems Problems to which we get answers: 1 Find the best exponent γ ( q , c ) := γ ( c ) for which ∥ σ ( c ) N ∥ ∞ = O ( N γ ( c ) ) . 2 (Multifractally ) analyze the size of the sets { x ∈ R : | σ ( c ) q n ( x ) | ∼ q n α } , 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 10 / 32

Known facts Known facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 11 / 32

Known facts About γ ( q , c ), c ̸ = 0 1 2 ≤ γ ( q , c ) < 1. Mauduit-Rivat-Sarkozy (2017): γ (2 , c ) ≤ π 2 ∥ c ∥ 20 log 2 . Fan-Koniezny (2018): Gowers uniform norms Assume f : N → { z ∈ C , | z | = 1 } be q -multiplicative.Then ∥ f ∥ U s ≪ ∥ f ∥ τ s ∀ s ≥ 2 , ∃ τ s > 0 , U 1 . So, γ ( q , c ) < 1 implies: ∀ P ∈ R [ x ], ∃ γ ′ < 1. t ( q , c ) ∥ ∑ N − 1 e 2 π iP ( n ) x ∥ ∞ = O ( N γ ′ ). n 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 12 / 32

Known facts Subsequences of TM sequence Theorem (Mauduit-Rivat, 2010/2009) For any α ∈ (0 , 1) , there exists σ ( α ) > 0 such that e 2 π i α S 2 ( p ) ≪ N 1 − σ ( α ) . ∑ p ≤ N , p ∈P For any α ∈ (0 , 1) , there exists c ( α ) > 0 such that e 2 π i α S 2 ( n 2 ) ≪ N 1 − c ( α ) (1 + log log N ) 5 . ∑ n ≤ N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 13 / 32

Ergodic maximization Ergodic maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 14 / 32

Ergodic maximization Dynamical interpretation 1 1 1 e 2 π ic ∑ m − 1 j =0 a j e 2 π i ∑ m − 1 j =0 a j 2 j x . σ ( c ) ∑ ∑ ∑ 2 m ( x ) = · · · a 0 =0 a 1 =0 a m − 1 =0 m − 1 ( 1 + e 2 π i ( c +2 j x ) ) ∏ = j =0 m − 1 | σ ( c ) ∏ 2 m ( x ) | = 2 m | cos( π (2 j x + c )) | . j =0 Let f c ( x ) = log | cos( π ( x + c )) | and let T ( x ) = 2 x mod 1 . Then m − 1 1 2 m − 1 ( x ) | = log 2 + 1 m log | σ ( c ) ∑ f c ( T j ( x )) . m . . . . . . . . . . . . . . . . . . . . j =0 . . . . . . . . . . . . . . . . . . . . November 5, 2019 15 / 32

Ergodic maximization For q ≥ 2, we have Tx = qx mod 1 and � � sin π q ( x + c ) � � f c ( x ) = log � . � � q sin π ( x + c ) � Figure: The graph of f 0 on the interval [ − 1 / q , 1 − 1 / q ], here q = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 16 / 32

Ergodic maximization Ergodic maximization Observation: f c is upper semi-continuous. M : the set of all T -invariant Borel probability measures on T = R / Z . We have {∫ } β ( c ) = sup f c d µ : µ ∈ M . By Birkhorff Ergodic Theorem, γ ( c ) ≥ 1 + β ( c ) log q . Fact. We actually have equality. (Conze-Guivarc’h unpublished , Jenkinson) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 17 / 32

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 fd µ : µ ∈ M T . M 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?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . November 5, 2019 18 / 32