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

TL;DR

This paper explores the properties of counting and mass dimensions for subsets of integer lattices, establishing Marstrand-type theorems for these dimensions and applying them to sumsets in Euclidean spaces.

Contribution

It introduces basic properties, characterizations, and Marstrand-type theorems for counting and mass dimensions in $\,\mathbb{Z}^d$, extending recent related work.

Findings

Marstrand-type theorems for counting and mass dimensions

Bounds on dimensions of sumsets for almost every coefficient vector

Characterization of counting dimension via coverings

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 $\min \big(k,D(A)\big)$. As an application, for subsets $A_1, \ldots, A_d$ 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.