D sets and IP rich sets in \(\mathbb {Z}\) (Q2787132)

The operation \(+\) on \(\mathbb Z\) has a unique associative extension to \(\beta\mathbb Z\) with the property that for every \(q\in\beta\mathbb Z\) the function \(p\mapsto p+q\) is continuous. With this operation the Stone-Čech compactification \(\beta\mathbb Z\) of \(\mathbb Z\) is a compact right topological semigroup. An ultrafilter \(p\in\beta\mathbb Z\) is idempotent if \(p+p=p\); \(p\) is said to be essential if for every set \(A\in p\) the upper Banach density \(d^\ast (A)=\limsup_{d\to\infty}| A\cap[n+1,n+d]| /d\) is positive. A set \(A\subset\mathbb Z\) is said to be an IP set if it belongs to an idempotent ultrafilter; \(A\) is an IP\(^\ast\) set if it belongs to every idempotent ultrafilter; \(A\) is a D set if it belongs to an essential idempotent ultrafilter; \(A\) is a D\(^\ast\) set if it belongs to every essential idempotent ultrafilter; \(A\) is an AIP set (or IP rich), if \(A\setminus E\) is IP for every \(E\subset\mathbb Z\) with \(d^\ast (E)=0\); \(A\) is an AIP\(^\ast\) set, if \(A\cup E\) is IP\(^\ast\) for some \(E\subset\mathbb Z\) with \(d^\ast (E)=0\). Some relations between these notions are known and it was a question of V. Bergelson whether every D\(^\ast\) set is AIP\(^\ast\). In the paper under review the authors answers this question in the negative. This yields the following proper inclusions: IP\(^\ast\subsetneq\) AIP\(^\ast \subsetneq\) D\(^\ast\) and D\(\subsetneq\) AIP \(\subsetneq\) IP. Useful combinatorial characterizations of AIP sets and D sets are obtained.