Spacings and pair correlations for finite Bernoulli convolutions

Details Spacings and pair correlations for finite Bernoulli convolutions Spacings and pair correlations for finite Bernoulli convolutions

2008-08-11

arxiv.org/abs/0808.1568v1

Nonlinearity 22 (2009), no. 2, 381-393

Mathematics

Number Theory

17 pages, 6 figures

Scientific paper

10.1088/0951-7715/22/2/008

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.

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