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 \lambda\ the counting dimension of E+[\lambda F] 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+[\lambda F] has positive upper Banach density for Lebesgue almost every \lambda. The result has direct consequences when E,F are arithmetic sets, e.g. the integer values of a polynomial with integer coefficients.