Marstrand-type theorems for the counting and mass dimensions in ${\mathbb{Z}}^d$

Abstract

The counting and (upper) mass dimensions are notions of dimension for subsets of ${\mathbb{Z}}^d$. We develop their basic properties and give a characterization of the counting dimension via coverings. In addition, we prove Marstrand-type results for both dimensions. For example, if $A\subseteq {\mathbb{R}}^d$ has counting dimension $D(A)$, then for almost every orthogonal projection with range of dimension $k$, the counting dimension of the image of $A$ is at least $\min (k,D(A))$. As an application, for subsets $A_1,\dots ,A_d$ of $\mathbb{R}$, we are able to give bounds on the counting and mass dimensions of the sumset $c_1{A}_1+\cdots +c_d{A}_d$ for Lebesgue-almost every $c\in {\mathbb{R}}^d$. This work extends recent work of Y. Lima and C. G. Moreira.

41 pages

References in corpus (1)

• Theory of dimension for large discrete sets and applications

Cited by in corpus (5)

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