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 EN ∈ {−1, 1}N in order to measure their ‘level of randomness’. Those parameters, the normality measure N(EN), the well-distribution measure W(EN), and the correlation measure Ck(EN) of order k, focus on different combinatorial aspects of EN. In their work, amongst others, Mauduit and Sárközy (i) investigated the relationship among those parameters and their minimal possible value, (ii) estimated N(EN), W(EN), and Ck(EN) for certain explicitly constructed sequences EN suggested to have a ‘pseudorandom nature’, and (iii) investigated the value of those parameters for genuinely random sequences EN.

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(EN), W(EN), and Ck(EN) for typical EN. We prove that, for most EN ∈ {−1, 1}N, both W(EN) and N(EN) are of order √ N, while Ck(EN) is of order

• N log

N

k

• for any given 2 ≤ k ≤ N/4.

These results were improved later by Cristoph Aistleitner, who proved the existence of limit distributions for W(EN)/ √ N and N(EN)/ √ N, and by Kai-Uwe Schmidt, who proved the existence of (constant) limit distributions for Ck(EN)/

• N log

N

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 Ck(EN) (k even) for

arbitrary sequences EN, establishing one of the conjectures by Cassaigne, Mauduit and Sárközy. We also give an algebraic construction for a sequence EN with normality measure N(EN) < N1/3+o(1). This was later improved by Cristoph Aistleitner, who proved that the minimum value of N(EN) is O(log2(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(p1 . . . pk) = (−1)k if p1, . . . , pk 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(2iπx). 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(3n) > 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(3n) > 0.

This conjecture was proved in 1969 by Newman. Later, Coquet gave a precise formula for

n<N t(3n) : for every

integer N ≥ 1, we have

• h<N

t(3n) = NαF(log4 N) + O(1), (1) where F is a continuous nowhere differentiable periodic function of period 1 with inf F = 2 √ 3 3 , sup F = 55 3 ( 3 65)α (2)

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

Let us define, for every integer N ≥ 1 S(N) =

• n<N

n∈Q

t(n) =

• n<N

µ2(n)t(n). (3) The following result with Christian, which implies that S(N) is strictly negative for every large enough integer N is the following Theorem We have S(N) = − 2

π2 (1 + o(1))

N

3

α F

• log4

N 3

• , where F is the

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 ∈ AN, pw(n) ≤ f(n), ∀n ∈ N} and we denote Ln(f) =

• w∈W(f)

Ln(w).

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) = w0w1 · · · wi · · · (4)

• n the alphabet A, where x =

i≥0 wi qi+1 is the representation in

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 C(f) = {x =

• i≥0

wi qi+1 ∈ [0, 1], w(x) = w0w1 · · · wi · · · ∈ W(f)} (5)

• f 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), Ln(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 EW(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 EW(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 EW(f) can be strictly smaller than the limiting lower exponential growth rate E0(f) = lim

n→∞ inf 1 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 EW(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) = EW(f) E0(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 EW(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

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

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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! (. . . )