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A Marstrand Theorem for Subsets of Integers

Published online by Cambridge University Press:  25 October 2013

YURI LIMA  and
CARLOS GUSTAVO MOREIRA
YURI LIMA
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: yurilima@gmail.com)
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brasil (e-mail: gugu@impa.br)
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Abstract

We propose a counting dimension for subsets of $\mathbb{Z}$ and prove that, under certain conditions on E,F ⊂ $\mathbb{Z}$, for Lebesgue almost every λ ∈ $\mathbb{R}$ 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 ∈ $\mathbb{R}$. The result has direct consequences when E,F are arithmetic sets, e.g., the integer values of a polynomial with integer coefficients.

Keywords

Primary 05A99Secondary 37E15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 116 - 134
Copyright
Copyright © Cambridge University Press 2013 

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