Translation Properties of Sets of Positive Upper Density

Abstract

Generalizing a result of Raimi we show that there exists a set $E \subset {\mathbf {N}}$ such that if $A \subset {\mathbf {N}}$ is a set with positive upper density, then there exists a number $k \in {\mathbf {N}}$ such that ${d^ * }((A + k) \cap E) > 0$ and ${d^ * }((A + k) \cap {E^c}) > 0$. Some extensions and further results are also obtained.