A description of the segment \([1,T]\), where \(T\) is the meeting number of a set-lattice
Let \(L\) be a point lattice in \(\mathbb{R}^ n\), not necessarily of dimension \(n\), and let \(A \subset \mathbb{R}^ n\). The problem to estimate the number \(T\) of lattice translates \(A+u\), \(u \in L\), which meet \(A\) has been investigated intensively in the geometry of numbers [see e.g. \textit{P. Erdős} and \textit{C. A. Rogers}, Acta Arith. 9, 41-45 (1964; Zbl 0132.032) or \textit{H. Groemer}, Monatsh. Math. 74, 21-29 (1970; Zbl 0199.571)]. In an earlier paper [in: B. Riemann Memorial Volume, World Scientific (to appear)] the author has proved the inequality \[ T \geq \bigl( d(x) + 1 \bigr) \# \bigl( (A-x) \cap L \bigr) - d(x) \bigl( d(x) + 1 \big)/2 \] for any \(x \in \mathbb{R}^ n\) where \(d(x)\) is the dimension of the affine hull of \((A - x) \cap L\). By clever arguments the author further improves this estimate. His bounds are in some sense best possible.
scientific article; zbMATH DE number 508819
scientific article