On the pair correlations of powers of real numbers

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Abstract

A classical theorem of Koksma states that for Lebesgue almost every x > 1 the sequence (x n) n=1 is uniformly distributed modulo one. In the present paper we extend Koksma’s theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every x > 1 the pair correlations of the fractional parts of (x n) n=1 are asymptotically Poissonian. The proof is based on a martingale approximation method.

Original languageEnglish
Pages (from-to)243-268
Number of pages26
JournalIsrael Journal of Mathematics
Volume242
Issue number1
DOIs
Publication statusPublished - Apr 2021

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

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