Sequences on finite alphabets: my collaboration with Christian - PowerPoint PPT Presentation

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Sequences on finite alphabets: my collaboration with Christian Carlos Gustavo Tamm de Araujo Moreira (Gugu) (IMPA, Rio de Janeiro, Brasil) Prime Numbers, Determinism and Pseudorandomness - CIRM - Luminy - Marseille - 07/11/2019

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences E N ∈ {− 1 , 1 } N in order to measure their ‘level of randomness’. Those parameters, the normality measure N ( E N ) , the well-distribution measure W ( E N ) , and the correlation measure C k ( E N ) of order k , focus on different combinatorial aspects of E N . In their work, amongst others, Mauduit and Sárközy ( i ) investigated the relationship among those parameters and their minimal possible value, ( ii ) estimated N ( E N ) , W ( E N ) , and C k ( E N ) for certain explicitly constructed sequences E N suggested to have a ‘pseudorandom nature’, and ( iii ) investigated the value of those parameters for genuinely random sequences E N .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences In collaboration with Christian, Yoshi Kohayakawa, Vojta Rödl and Noga Alon, we continue the work in the direction of ( iii ) above and determine the order of magnitude of N ( E N ) , W ( E N ) , and C k ( E N ) for typical E N . We prove that, for √ most E N ∈ {− 1 , 1 } N , both W ( E N ) and N ( E N ) are of order N , � � N � for any given 2 ≤ k ≤ N / 4. while C k ( E N ) is of order N log k These results were improved later by Cristoph Aistleitner, who √ proved the existence of limit distributions for W ( E N ) / N and √ N ( E N ) / N , and by Kai-Uwe Schmidt, who proved the existence of (constant) limit distributions for � � N � C k ( E N ) / N log . k

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences In another paper, we prove a lower bound (of the order of � N / k ) for the correlation measure C k ( E N ) ( k even) for arbitrary sequences E N , establishing one of the conjectures by Cassaigne, Mauduit and Sárközy. We also give an algebraic construction for a sequence E N with normality measure N ( E N ) < N 1 / 3 + o ( 1 ) . This was later improved by Cristoph Aistleitner, who proved that the minimum value of N ( E N ) is O (log 2 ( N )) .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Definitions: For each positive integer n , let s ( n ) be the number of bits equal to 1 in the binary representation of n (in other words, the sum of digits of n written in basis 2). We denote by µ the Möbius function defined by µ ( 1 ) = 1 , µ ( p 1 . . . p k ) = ( − 1 ) k if p 1 , . . . , p k are distinct prime numbers and µ ( n ) = 0 if n is divisible by the square of a prime number. We put α = log 3 log 4 = 0 , 7924812503605780907268694719739 ... For every real number x , e ( x ) = exp( 2 i π x ) . The Morse-Thue sequence is defined as t = ( t ( n )) n ∈ N = (( − 1 ) s ( n ) ) n ∈ N .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Leo Moser formulated in the sixties the following conjecture: For every N ≥ 1, we have � n < N t ( 3 n ) > 0.

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Leo Moser formulated in the sixties the following conjecture: For every N ≥ 1, we have � n < N t ( 3 n ) > 0. This conjecture was proved in 1969 by Newman. Later, Coquet gave a precise formula for � n < N t ( 3 n ) : for every integer N ≥ 1, we have � t ( 3 n ) = N α F (log 4 N ) + O ( 1 ) , (1) h < N where F is a continuous nowhere differentiable periodic function of period 1 with √ inf F = 2 3 sup F = 55 3 ( 3 65 ) α , (2) 3

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Let us define, for every integer N ≥ 1 � � µ 2 ( n ) t ( n ) . S ( N ) = t ( n ) = (3) n < N n < N n ∈ Q The following result with Christian, which implies that S ( N ) is strictly negative for every large enough integer N is the following Theorem � N � α F We have S ( N ) = − 2 N � � π 2 ( 1 + o ( 1 )) log 4 , where F is the 3 3 Coquet function.

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences The complexity function of an infinite word w on a finite alphabet A is the sequence counting, for each non-negative n , the number of words of length n on the alphabet A that are factors of the infinite word w . Our work concerns the study of infinite sequences w the complexity function of which is bounded by a given function f from N to R + . More precisely, if f is such a function, we consider the set W ( f ) = { w ∈ A N , p w ( n ) ≤ f ( n ) , ∀ n ∈ N } and we denote � L n ( f ) = L n ( w ) . w ∈ W ( f )

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences To each x ∈ [ 0 , 1 ] we associate the infinite word w ( x ) = w 0 w 1 · · · w i · · · (4) w i on the alphabet A , where x = � q i + 1 is the representation in i ≥ 0 base q of the real number x (when x is a q -adic rational number, we choose for x the infinite word ending with 0 ∞ ). We define w i � C ( f ) = { x = q i + 1 ∈ [ 0 , 1 ] , w ( x ) = w 0 w 1 · · · w i · · · ∈ W ( f ) } i ≥ 0 (5) of real numbers x ∈ [ 0 , 1 ] the q − adic expansion of which has a complexity function bounded by f .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences When f has sub-exponential growth, we give upper and lower estimates “of the same type" (except in the case of linear growth) for the sizes of W ( f ) , L n ( f ) and for the generalized Hausforff dimensions of C ( f ) . In the case when f has exponential growth, we introduce a real parameter, the word entropy E W ( f ) associated to a given function f and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by f in terms of its word entropy. We presented a combinatorial proof of the fact that E W ( f ) is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by f and we give several examples showing that even under strong conditions on f , the word entropy E W ( f ) can be strictly smaller than the limiting n →∞ inf 1 lower exponential growth rate E 0 ( f ) = lim n log f ( n ) of f .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences We also gave estimates on the word entropy E W ( f ) in terms of the limiting lower exponential growth rate of f . Assume that f is good , i.e., it satisfies some natural conditions, which are satisfied for complexity functions (as f ( m + n ) ≤ f ( m ) f ( n ) ). We call entropy ratio of f the quantity ρ ( f ) = E W ( f ) E 0 ( f ) . Then, inf { ρ ( f ) , f is good } = 1 2 .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences In collaboration with Sébastien, we gave an algorithm to estimate with arbitrary precision E W ( f ) from finitely many values of f (for good functions f ). In general, its complexity if very large (like a tower), and involves the construction of full subshifts of finite type contained in W ( f ) .

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences ]

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences ]

Measures of pseudorandomness of finite binary sequences Moser-Newman phenomenon for square-free numbers Infinite sequences Muito obrigado! Muchas gracias! Thank you very much! Merci beaucoup! ( . . . )