On the correlations of nα mod 1

Niclas Technau; Nadav Yesha

doi:10.4171/JEMS/1281

On the correlations of n^α mod 1

Niclas Technau, Nadav Yesha

Department of Mathematics

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

Abstract

A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts ¹nαºof the sequence (nα)_n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence (¹nαº)_n≥1 are of fundamental importance. We prove that for each k ≥ 2; the k-level correlation function R_k is Poissonian for almost every α > 4k² - 4k - 1.

Original language	English
Pages (from-to)	4123-4154
Number of pages	32
Journal	Journal of the European Mathematical Society
Volume	25
Issue number	10
DOIs	https://doi.org/10.4171/JEMS/1281
State	Published - 2023

Bibliographical note

Publisher Copyright:
© 2022 European Mathematical Society Published by EMS Press and licensed under a CC BY 4.0 license.

Keywords

Uniform distribution modulo 1
correlation functions
local statistics

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.4171/JEMS/1281

Cite this

@article{1262cbc25bc74ce7a94a5d51529f6df8,

title = "On the correlations of nα mod 1",

abstract = "A well known result in the theory of uniform distribution modulo 1 (which goes back to Fej{\'e}r and Csillag) states that the fractional parts 1nαºof the sequence (nα)n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence (1nαº)n≥1 are of fundamental importance. We prove that for each k ≥ 2; the k-level correlation function Rk is Poissonian for almost every α > 4k2 - 4k - 1.",

keywords = "Uniform distribution modulo 1, correlation functions, local statistics",

author = "Niclas Technau and Nadav Yesha",

note = "Publisher Copyright: {\textcopyright} 2022 European Mathematical Society Published by EMS Press and licensed under a CC BY 4.0 license.",

year = "2023",

doi = "10.4171/JEMS/1281",

language = "English",

volume = "25",

pages = "4123--4154",

journal = "Journal of the European Mathematical Society",

issn = "1435-9855",

publisher = "European Mathematical Society Publishing House",

number = "10",

}

TY - JOUR

T1 - On the correlations of nα mod 1

AU - Technau, Niclas

AU - Yesha, Nadav

PY - 2023

Y1 - 2023

N2 - A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts 1nαºof the sequence (nα)n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence (1nαº)n≥1 are of fundamental importance. We prove that for each k ≥ 2; the k-level correlation function Rk is Poissonian for almost every α > 4k2 - 4k - 1.

AB - A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts 1nαºof the sequence (nα)n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence (1nαº)n≥1 are of fundamental importance. We prove that for each k ≥ 2; the k-level correlation function Rk is Poissonian for almost every α > 4k2 - 4k - 1.

KW - Uniform distribution modulo 1

KW - correlation functions

KW - local statistics

UR - http://www.scopus.com/inward/record.url?scp=85173940779&partnerID=8YFLogxK

U2 - 10.4171/JEMS/1281

DO - 10.4171/JEMS/1281

M3 - Article

AN - SCOPUS:85173940779

SN - 1435-9855

VL - 25

SP - 4123

EP - 4154

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 10

ER -