Intermediate-scale statistics for real-valued lacunary sequences

Nadav Yesha

doi:10.1017/S0305004123000142

Intermediate-scale statistics for real-valued lacunary sequences

Nadav Yesha

Department of Mathematics

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

Abstract

We study intermediate-scale statistics for the fractional parts of the sequence, where is a positive, real-valued lacunary sequence, and. In particular, we consider the number of elements in a random interval of length, where, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t., which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in when. For slowly growing L, we further prove a central limit theorem for which holds for almost all.

Original language	English
Pages (from-to)	303-318
Number of pages	16
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	175
Issue number	2
DOIs	https://doi.org/10.1017/S0305004123000142
State	Published - 11 Sep 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society.

Keywords

11K06 11K99

ASJC Scopus subject areas

General Mathematics

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10.1017/S0305004123000142

Cite this

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abstract = "We study intermediate-scale statistics for the fractional parts of the sequence, where is a positive, real-valued lacunary sequence, and. In particular, we consider the number of elements in a random interval of length, where, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t., which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in when. For slowly growing L, we further prove a central limit theorem for which holds for almost all.",

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AB - We study intermediate-scale statistics for the fractional parts of the sequence, where is a positive, real-valued lacunary sequence, and. In particular, we consider the number of elements in a random interval of length, where, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t., which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in when. For slowly growing L, we further prove a central limit theorem for which holds for almost all.

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Intermediate-scale statistics for real-valued lacunary sequences

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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