Deviations of Ergodic sums for Toral Translations I : Convex bodies

TL;DR

This paper establishes the existence of a limiting distribution for normalized discrepancy functions of random toral translations relative to convex bodies, linking Fourier analysis, lattice dynamics, and geometric counting problems.

Contribution

It introduces a new limiting distribution for discrepancy functions in toral translations and connects it to lattice dynamics and geometric counting problems.

Findings

Existence of a limiting distribution for discrepancy functions.

Identification of the distribution with level sets of a function on lattice space and tori.

Applications to lattice point counting and geodesic time in flat tori.

Abstract

We show the existence of a limiting distribution $\cD_\cC$ of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set $\cC$. 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 $\cD_\cC$ 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.