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A Marstrand Theorem for Subsets of Integers
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- Journal:
- Combinatorics, Probability and Computing / Volume 23 / Issue 1 / January 2014
- Published online by Cambridge University Press:
- 25 October 2013, pp. 116-134
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- Article
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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.
