A Marstrand theorem for subsets of integers

Details A Marstrand theorem for subsets of integers A Marstrand theorem for subsets of integers

2010-11-02

arxiv.org/abs/1011.0672v2

Mathematics

Dynamical Systems

19 pages, no figures, two sections added

Scientific paper

We prove a Marstrand type theorem for a class of subsets of the integers. More specifically, after defining the counting dimension D(E) of subsets of Z and the concepts of regularity and compatibility, we show that if E,F are two regular compatible subsets of Z, then D(E+[cF]) is at least min{1,D(E)+D(F)} for Lebesgue almost every real number c. If in addition D(E)+D(F)>1, then E+[cF] has positive upper-Banach density for Lebesgue almost every c. The result has direct consequences when applied to arithmetic sets, such as the integer values of a polynomial with integer coefficients.

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