Measure of Self-Affine Sets and Associated Densities

TL;DR

This paper investigates the measure-theoretic properties of self-affine sets generated by expanding matrices and digit sets, establishing connections between Lebesgue measure, Hausdorff measure, and associated densities.

Contribution

It provides new relationships between the Lebesgue measure of self-affine sets and the upper Beurling density, as well as linking Hausdorff measure to upper density for similarity matrices.

Findings

Lebesgue measure of self-affine sets relates to upper Beurling density when digit set size equals determinant.

Hausdorff measure of self-affine sets relates to a density notion when digit set size is less than determinant and matrix is a similarity.

Established formulas connect measure and density for self-affine sets with different digit set sizes.

Abstract

Let $B$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The self-affine set $K=K(B,\mathcal{D})$ is the unique compact set satisfying the set-valued equation $BK=\displaystyle\bigcup_{d\in\mathcal{D}}(K+d)$. In the case where $\text{card}(\mathcal{D})=\lvert\det B\rvert,$ we relate the Lebesgue measure of $K(B,\mathcal{D})$ to the upper Beurling density of the associated measure $\mu=\lim\limits_{s\to\infty}\sum\limits_{\ell_0,\dotsc,\ell_{s-1}\in\mathcal{D}}\delta_{\ell_0+B\ell_1+\dotsb+B^{s-1}\ell_{s-1}}.$ If, on the other hand, $\text{card}(\mathcal{D})<\lvert\det B\rvert$ and $B$ is a similarity matrix, we relate the Hausdorff measure $\mathcal{H}^s(K)$, where $s$ is the similarity dimension of $K$, to a corresponding notion of upper density for the measure $\mu$.