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arXiv · 1306.0028

The distribution of directions in an affine lattice: two-point correlations and mixed moments

Daniel El-Baz·Jens Marklof·Ilya Vinogradov
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Abstract

We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most $T$. Str\"ombergsson and the second author proved in [Annals of Math.~173 (2010), 1949--2033] that the distribution of gaps between the lattice directions has a limit as $T$ tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs distinctly from the gap distribution observed in a Poisson process, which is exponential. The present study shows that the limiting two-point correlation function of the projected lattice points exists and is Poissonian. This answers a recent question by Boca, Popa and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments.

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Daniel El-Baz, Jens Marklof, Ilya Vinogradov. 2013-05-31. The distribution of directions in an affine lattice: two-point correlations and mixed moments. https://arxiv.org/abs/1306.0028

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