Measures of Pseudorandomness for Finite Sequences: Typical Values

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences EN ∈ {−1, 1} 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} , both W (EN ) and N (EN ) are of order √ N , while Ck(EN ) is of order q N log ` N k ́ for any given 2 ≤ k ≤ N/4. Date: Copy produced on April 12, 2006. 1991 Mathematics Subject Classification. 68R15.