A Marstrand theorem for subsets of integers

Abstract

We propose a counting dimension for subsets of Z and prove that, under certain conditions on two such subsets E and F, for Lebesgue almost every real λ the counting dimension of E+[λF] is at least the minimum between 1 and the sum of the counting dimensions of E and F. Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E+[λF] has positive upper Banach density for Lebesgue almost every λ. The result has direct consequences when E,F are arithmetic sets, e.g. the integer values of a polynomial with integer coefficients.

arXiv comment16 pages, to appear in Combinatorics, Probability and Computing

References in corpus (2)

• A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets
• Yet another proof of Marstrand's Theorem

Cited by in corpus (6)

• Marstrand-type theorems for the counting and mass dimensions in ${\mathbb{Z}}^d$
• A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets
• Additive and geometric transversality of fractal sets in the integers
• A combinatorial proof of a sumset conjecture of Furstenberg
• A Marstrand type slicing theorem for subsets of ${\mathbb{Z}}^2\subset {\mathbb{R}}^2$ with the mass dimension
• Marstrand type slicing statements in ${\mathbb{Z}}^2\subset {\mathbb{R}}^2$ are false for the counting dimension