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arXiv · 1310.5729

High density piecewise syndeticity of sumsets

Mauro Di Nasso·Isaac Goldbring·Renling Jin·Steven Leth·Martino Lupini·Karl Mahlburg
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Abstract

Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A+B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B, there is a high density set of witnesses to the piecewise syndeticity of A+B. Most of the result are shown to hold more generally for subsets of Z^d. The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

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Mauro Di Nasso, Isaac Goldbring, Renling Jin, Steven Leth, Martino Lupini, Karl Mahlburg. 2013-10-28. High density piecewise syndeticity of sumsets. https://doi.org/10.1016/j.aim.2015.03.009

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