A Marstrand theorem for subsets of integers

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.

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