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- W2515980152 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 { C i } 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." @default.
- W2515980152 created "2016-09-16" @default.
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- W2515980152 date "2016-03-16" @default.
- W2515980152 modified "2023-10-15" @default.
- W2515980152 title "Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ" @default.
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- W2515980152 doi "https://doi.org/10.1017/s096354831600002x" @default.
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