Gap statistics and higher correlations for geometric progressions modulo one

TL;DR

This paper proves that for almost every real number greater than one, the fractional parts of its geometric progression exhibit Poissonian gap distributions and correlations, extending classical equidistribution results with an analytic approach.

Contribution

The authors sharpen Koksma's equidistribution theorem by establishing Poissonian limit distributions for correlations and gaps of geometric progressions modulo one using analytic methods.

Findings

Correlations of all finite orders have Poissonian limit distributions.

Normalized gaps of geometric progressions mod 1 are Poissonian.

Results extend to certain sub-lacunary sequences.

Abstract

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $\alpha>1$, the fractional parts of the geometric progression $(\alpha^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $\alpha>1$, the correlations of all finite orders and hence the normalized gaps of $(\alpha^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, 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.