Measures of pseudorandomness for finite sequences: Typical values

N. Alon; Y. Kohayakawa; C. Mauduit; C. G. Moreira; V. Rödl

doi:10.1112/plms/pdm027

Measures of pseudorandomness for finite sequences: Typical values

N. Alon
, Y. Kohayakawa
, C. Mauduit
, C. G. Moreira
, V. Rödl

Research output: Contribution to journal › Article › peer-review

53 Scopus citations

Abstract

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 (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 (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 en.In this paper, we continue the work in the direction of (iii) above and determine the order of magnitude of (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 (E_N) are of order √ N, while C _k(E_N) is of order Nlog(Nk) for any givE_N 2 ≤ k ≤ N/4.

Original language	English
Pages (from-to)	778-812
Number of pages	35
Journal	Proceedings of the London Mathematical Society
Volume	95
Issue number	3
DOIs	https://doi.org/10.1112/plms/pdm027
State	Published - 1 Jan 2007
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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title = "Measures of pseudorandomness for finite sequences: Typical values",

abstract = "Mauduit and S{\'a}rk{\"o}zy 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 (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 S{\'a}rk{\"o}zy (i) investigated the relationship among those parameters and their minimal possible value, (ii) estimated (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 (E N), W(EN) and Ck(EN) for typical EN. We prove that, for most EN ∈ [-1, 1]N, both W(EN) and (EN) are of order √ N, while C k(EN) is of order Nlog(Nk) for any givEN 2 ≤ k ≤ N/4.",

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T2 - Typical values

AU - Alon, N.

AU - Kohayakawa, Y.

AU - Mauduit, C.

AU - Moreira, C. G.

AU - Rödl, V.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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 (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 (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 (E N), W(EN) and Ck(EN) for typical EN. We prove that, for most EN ∈ [-1, 1]N, both W(EN) and (EN) are of order √ N, while C k(EN) is of order Nlog(Nk) for any givEN 2 ≤ k ≤ N/4.

AB - 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 (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 (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 (E N), W(EN) and Ck(EN) for typical EN. We prove that, for most EN ∈ [-1, 1]N, both W(EN) and (EN) are of order √ N, while C k(EN) is of order Nlog(Nk) for any givEN 2 ≤ k ≤ N/4.

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JF - Proceedings of the London Mathematical Society

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Measures of pseudorandomness for finite sequences: Typical values

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