Heaviness in Toral Rotations

TL;DR

This paper studies the size of the set of points in a torus whose orbits under rotation frequently hit a fixed target more often than expected, providing bounds based on geometric and number-theoretic properties.

Contribution

It introduces bounds on the Minkowski dimension of the strictly heavy set in toral rotations, extending results to general compact abelian groups like p-adic integers.

Findings

Upper bounds for the Minkowski dimension of heavy sets.

Extension of results to compact abelian groups beyond tori.

Connections between orbit hitting frequency and Diophantine approximation.

Abstract

We investigate the dimension of the set of points in the d-torus which have the property that their orbit under rotation by some alpha hits a fixed closed target A more often than expected for all finite initial portions. An upper bound for the lower Minkowski dimension of this "strictly heavy set" H(A,alpha) is found in terms of the upper Minkowski dimension of the boundary of A, as well as k, the Diophantine approximability from below of the Lebesgue measure of A. The proof extends to translations in compact abelian groups more generally than just the torus, most notably the p-adic integers.