Lattices modulo \(N\) with long shortest distances
For coprime integers \(a\), \(b\) with \(0<a,b<N\) let \(\Pi_{N,a,b}\) be a set \(\{(na \pmod N, nb\pmod N):0\leq n < N \}\). \(\Pi_{N,a,b}\) can be considered as a lattice modulo \(N\). Denote by \(\lambda_1(\Pi_{N,a,b})\) the shortest distance between points in \(\Pi_{N,a,b}\). Further, suppose \(f_{\max}(N)=\max_{1\leq a,b < N}\) \(\lambda_1(\Pi_{N,a,b})/\sqrt{N}\). Then it is proved that for any \(\varepsilon>0\) there exist infinitely many \(N\) such that \(f_{\max}(N)>\sqrt{\frac{2}{\sqrt{3}}}-\varepsilon\). As a corollary, it is proved that for any \(\varepsilon>0\) there exists an integer matrix \(X\) such that \(\Delta(\Lambda_h)-\Delta(X\mathbb Z^2)<\varepsilon\), where \(\Lambda_h\) is a hexagonal lattice and \(\Delta(\cdot)\) is a lattice packing density.
scientific article; zbMATH DE number 6808303