Journal
GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 24, Issue 1, Pages 85-115Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00039-014-0254-y
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Funding
- NSF
- project ANR GeoDyM
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1101635] Funding Source: National Science Foundation
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We show the existence of a limiting distribution of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set . Using a correspondence between the small divisors in the Fourier series of the discrepancy function and lattices with short vectors, and mixing of diagonal flows on the space of lattices, we identify with the distribution of the level sets of a function defined on the product of the space of lattices with an infinite dimensional torus. We apply our results to counting lattice points in slanted cylinders and to time spent in a given ball by a random geodesic on the flat torus.
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