Pair correlation densities of inhomogeneous quadratic forms

2002/10/31 by Jens Marklof · 2 citations
Mathematics · Physics and Astronomy · #math.NT #math-ph #math.DS #math.MP #nlin.CD

paper · pdf

published as Ann. of Math. (2), Vol. 158 (2003), no. 2, 419--471 · 53 pages published version

arxiv created 2004/04/20 · arxiv updated 2009/11/30

Abstract

Under explicit diophantine conditions on (α,β)∈\RR², we prove that the local two-point correlations of the sequence given by the values (m-α)²+\break (n-β)², with (m,n)∈\ZZ², are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner's classification of measures invariant under unipotent flows.

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• A survey and a result on inhomogeneous quadratic forms
• Distribution of Farey fractions with k-free denominators

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