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The distribution of spacings between the fractional parts of $n^2 α$

Zeev Rudnick·Peter Sarnak·Alexandru Zaharescu
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Abstract

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.

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Zeev Rudnick, Peter Sarnak, Alexandru Zaharescu. 2001-01-04. The distribution of spacings between the fractional parts of $n^2 α$. https://doi.org/10.1007/s002220100141

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