High density piecewise syndeticity of sumsets

Renling Jin proved that if A and B   are two subsets of the natural numbers with positive Banach density, then A+BA+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+BA+B. Most of the results are shown to hold more generally for subsets of ZdZd. The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.