The set of neighbours of a set in a point-lattice (Q1336707)

For a lattice \(\Lambda \subset \mathbb{R}^ n\) and a bounded \(A \subset \mathbb{R}^ n\) the set \(S(A) = \{u \in \Lambda \mid A \cap (A + u) \neq \emptyset\}\) is called the set of neighbours of \(A\) in \(\Lambda\). Further \(N(A) = \text{card } S(A)\) is the number of neighbours (`Nachbarnzahl'), introduced by Hadwiger. The author improves lower bounds for \(N(A)\), given by Erdős, Rogers and Groemer.