A Marstrand theorem for subsets of integers
Dynamical Systems
2020-04-21 v3 Combinatorics
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.
Cite
@article{arxiv.1011.0672,
title = {A Marstrand theorem for subsets of integers},
author = {Yuri Lima and Carlos Gustavo Moreira},
journal= {arXiv preprint arXiv:1011.0672},
year = {2020}
}
Comments
16 pages, to appear in Combinatorics, Probability and Computing