Hausdorff dimension of affine random covering sets in torus

TL;DR

This paper determines the almost sure Hausdorff dimension of random affine covering sets in a torus, using singular value analysis of the linear transformations involved.

Contribution

It provides a formula for the Hausdorff dimension of random covering sets in the torus, extending previous results to more general affine sets and using singular value decomposition.

Findings

Derived a dimension formula based on singular values.

Established the result for parallelepipeds and linear images with interior.

Confirmed the formula holds under decreasing singular value sequences.

Abstract

We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + \xi_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $\xi_n\in\mathbb T^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.