A Marstrand type slicing theorem for subsets of $\mathbb{Z}^2 \subset \mathbb{R}^2$ with the mass dimension

Abstract

We prove a Marstrand type slicing theorem for the subsets of the integer square lattice. This problem is the dual of the corresponding projection theorem, which was considered by Glasscock, and Lima and Moreira, with the mass and counting dimensions applied to subsets of Z d . In this paper, more generally we deal with a subset of the plane that is 1 separated, and the result for subsets of the integer lattice follow as a special case. We show that the natural slicing question in this setting is true with the mass dimension.Recently there have been two proofs of this celebrated conjecture [6, 9].(2) We could in principle consider an arbitrary point set in R 2 , but when we have some limit points in our set, then the dimension of the set goes to infinity. For 1-separated sets in the plane, the dimension is at most 2.

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“…In this case, for some large k 0 > 0, (1) for each k ≥ k 0 , we fill the annular region inside the cone between the heights 2 2 k+1 and 2 2 k + 2 2 k+1 with all the points belonging to Z 2 within the annular region. This example was already considered in [2]. This is a set of mass dimension 3/2 with each of the tubes within the cone having mass dimension 1/2.…”

“…In [2], for the mass dimension in our setting of 1 separated subsets in R 2 , with a Tchebysheff and Fubini type argument the slicing statement was first shown to be true in a weak asymptotic sense, and then it was also shown to be true for Lebesgue almost every slice. One then specializes to sets A, B ⊂ N and considers the dimension of the intersection of the broken line {(x, y) : y = ⌊ũx + ṽ⌋, ũ > 0} with the Cartesian product set A × B.…”

“…This is the strongest slicing statement we can make with the counting dimension. In [2] we show that the corresponding theorem is false, i.e in a weak sense, the Marstrand type slicing theorem holds true with the mass dimension.…”

“…The natural dual slicing statement with the mass dimension was recently shown to be true by the author [2]. When dealing with the slicing question with a 1 separated set in R 2 , it is natural to work with a width 1 tube t u,v which is explicitly described as:…”

“…An analogous version of Theorem 3 is also true for the mass dimension, cf. Example 2 in Section 3 of [2].…”