Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

Abstract

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.