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.
Bibliographical noteFunding Information:
CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945, P-34763 and Y-901. NT received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement no. 786758), and Austrian Science Fund (FWF) from project J 4464-N. NY is supported by the ISRAEL SCIENCE FOUNDATION (Grant no. 1881/20).
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
ASJC Scopus subject areas
- Mathematics (all)