Deviations of ergodic sums for toral translations II. Boxes

TL;DR

This paper investigates the distribution of visits of ergodic sums for toral translations to random boxes, revealing a convergence to a Cauchy distribution after normalization, using advanced probabilistic and dynamical systems techniques.

Contribution

It establishes the asymptotic distribution of visit discrepancies for toral translations to random boxes, extending previous results to higher dimensions and more general sets.

Findings

Discrepancy normalized by ln^d N converges to a Cauchy distribution.

Uses a Poisson limit theorem for Cartan actions on lattice spaces.

Provides new insights into ergodic sums for toral translations.

Abstract

We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.