TY - JOUR
T1 - Gap statistics and higher correlations for geometric progressions modulo one
AU - Aistleitner, Christoph
AU - Baker, Simon
AU - Technau, Niclas
AU - Yesha, Nadav
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023/2
Y1 - 2023/2
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85123823794&partnerID=8YFLogxK
U2 - 10.1007/s00208-022-02362-3
DO - 10.1007/s00208-022-02362-3
M3 - Article
AN - SCOPUS:85123823794
SN - 0025-5831
VL - 385
SP - 845
EP - 861
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -