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

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.