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

The counting and (upper) mass dimensions of a set A ⊆ Rᵈ are D (A) = \|₂\| | A C | \|C\|, DD (A) = | A [-, ) ᵈ | (2), 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 ⊆ Rᵈ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: D (A) = inf \ 0 {d₇^{} (A) = 0 \}, where d₇^{} (A) = ₑ ₀ \|₂\| \ ᵢ (\|Cᵢ\|{\|C\|) ^ \ | \ 1 \|Cᵢ\| r \|C\| \} in which the infimum is taken over cubic coverings C i of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ Rᵈ 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.