Yuri Lima, Carlos Gustavo Moreira
Published 2010-11-02, updated 2013-09-23Version 3
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.
Comments: 16 pages, to appear in Combinatorics, Probability and Computing
Multiple ergodic theorems for arithmetic sets
Marstrand type slicing statements in $\mathbb{Z}^{2}\subset \mathbb{R}^{2}$ are false for the counting dimension
Szemeredi-type theorems for subsets of locally compact abelian groups of positive upper Banach density
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