A remark to the paper of H. Hadwiger: ``Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl''
Let \(\Lambda \in {\mathbb{R}}^ n\) be a point lattice, \(A\subset {\mathbb{R}}^ n\) be a Lebesgue-measurable set and V be the Lebesgue measure. In the paper the inequality \[ (1)\quad \alpha (A)\cdot N(A)\leq V((A-A)\cap \Lambda;A)-V(A) \] is proved, where \(\alpha\) (A) is the measure of covering \({\mathbb{R}}^ n\) by lattice translates of A, N(A) is the number of neighbours of A and \(V((A-A)\cap \Lambda;A)\) is a sort of ``measure'' of A introduced by the author. This result improves the inequality of \textit{H. Hadwiger} [ibid. 73, 213-217 (1969; Zbl 0182.557)]. Moreover, the inequality (2)\ 2(V(A)-\(\alpha\) (A))\(\leq \alpha (A)\cdot N(A)\) is also proved as a consequence of an earlier result of the author [Period. Math. Hung. 11, 95-103 (1980; Zbl 0412.10023)].
number of lattice points in difference set
scientific article; zbMATH DE number 4017006