The distribution of spacings between fractional parts of lacunary sequences

Abstract

Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to understand the distribution of the spacings between elements of such sequences (normalized to have unit mean), especially for sequences of polynomials. In this note, we consider lacunary sequences, such as the sequence of of powers of 2 (that is a(n)= 2^n). We show that for almost every t, all local correlation functions of the sequence t*2^n are those of a random sequence (the Poisson model). Among other things, this implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.

arXiv comment21 pages

References in corpus (1)

• Level Spacings for Integrable Quantum Maps in Genus Zero

Cited by in corpus (12)

• There is no Khintchine threshold for metric pair correlations
• Gaps between logs
• A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
• Spacings and pair correlations for finite Bernoulli convolutions
• Gap statistics and higher correlations for geometric progressions modulo one
• On a random walk with memory and its relation to Markovian processes
• Pair correlation densities of inhomogeneous quadratic forms II
• A construction of a $\lambda$- Poisson generic sequence
• Intermediate-scale statistics for real-valued lacunary sequences
• Continuous Distribution Arising From the Three Gaps Theorem
• On Inhomogeneous Poissonian Pair Correlations
• On the smallest gap in a sequence with Poisson pair correlations