A Marstrand theorem for subsets of integers

TL;DR

This paper introduces a new counting dimension for subsets of integers and proves a Marstrand-type theorem relating the dimensions of sumsets involving real scalar multiples, with implications for arithmetic sets.

Contribution

It establishes a Marstrand theorem for counting dimensions of sumsets of integer subsets, extending geometric measure theory concepts to discrete settings.

Findings

For almost every real mbda, the counting dimension of E + mbda F meets a lower bound.

If the sum of counting dimensions exceeds 1, the sumset has positive upper Banach density.

Results apply to integer values of polynomials with integer coefficients.

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.