Spacings and pair correlations for finite Bernoulli convolutions

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.

17 pages, 6 figures

References in corpus (3)

• The distribution of spacings between the fractional parts of $n^2\alpha$
• The distribution of spacings between fractional parts of lacunary sequences
• Random Walk with Shrinking Steps

Cited by in corpus (2)

• On a random walk with memory and its relation to Markovian processes
• Bernoulli Convolutions and 1D Dynamics