Measures of pseudorandomness for finite sequences: typical values
N. Alon, Yoshiharu Kohayakawa, Christian Mauduit, Carlos Gustavo Moreira, Vojtch Rödl · Proceedings of the London Mathematical Society · 2007
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. In this paper, 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.