The counting and (upper) mass dimensions of a set A ⊆
$\mathbb{R}^d$
are
$$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$
where ⌊
A⌋ denotes the set of elements of
A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes
C ⊆
$\mathbb{R}^d$
with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings:
$$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$
where
$${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha
\ \bigg| \
1 \leq \|C_i\| \leq r \|C\| \biggr\}$$
in which the infimum is taken over cubic coverings {
Ci
} of
A ∩
C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of
A ⊆
$\mathbb{R}^d$
under orthogonal projections with range of dimension
k have counting dimension at least min(
k,
D(
A)); if we assume
D(
A) =
D
(
A), then the mass dimension of
A under the typical orthogonal projection is equal to min(
k,
D(
A)). This work extends recent work of Y. Lima and C. G. Moreira.