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arXiv · 0808.1568

Spacings and pair correlations for finite Bernoulli convolutions

Itai Benjamini·Boris Solomyak
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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 $ν_r$, 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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Itai Benjamini, Boris Solomyak. 2008-08-11. Spacings and pair correlations for finite Bernoulli convolutions. https://doi.org/10.1088/0951-7715%2F22%2F2%2F008

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