Gap statistics and higher correlations for geometric progressions modulo one

Christoph Aistleitner, Simon Baker, Niclas Technau, Nadav Yesha

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Koksma’s equidistribution theorem from 1935 states that for Lebesgue almost every α>1, the fractional parts of the geometric progression (αn)n≥1 are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every α>1, the correlations of all finite orders and hence the normalized gaps of (αn)n≥1 mod 1 converge to the Poissonian model, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.
Original languageEnglish
JournalMathematische Annalen
Early online date28 Jan 2022
Publication statusE-pub ahead of print - 28 Jan 2022

ASJC Scopus subject areas

  • Mathematics(all)


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