Spacings and pair correlations for finite Bernoulli convolutions

TL;DR

This paper investigates the spacing and pair correlation properties of finite Bernoulli convolutions, revealing typical Poissonian spacing distributions and contrasting behaviors for algebraic parameters.

Contribution

It provides partial results on the pair correlation behavior of finite Bernoulli convolutions and highlights differences for algebraic versus generic parameters.

Findings

Pair correlations do not show attraction or repulsion on average.

Numerical evidence suggests Poissonian spacing distribution for generic parameters.

Behavior differs significantly for algebraic parameters.

Abstract

We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.