### 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 (US) |
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Pages (from-to) | 778-812 |

Number of pages | 35 |

Journal | Proceedings of the London Mathematical Society |

Volume | 95 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2007 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Proceedings of the London Mathematical Society*,

*95*(3), 778-812. https://doi.org/10.1112/plms/pdm027