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Renling Jin proved that if A and B are two subsets of the natural numberswith positive Banach density, then A+B is piecewise syndetic. In this paper, weprove that, under various assumptions on positive lower or upper densities of Aand B, there is a high density set of witnesses to the piecewise syndeticity ofA+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 inducedby cuts in the nonstandard integers.

High density piecewise syndeticity of sumsets