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