Covering sets for plane lattices (Q1915338)

Let \(L\) be a lattice and let \(K\) be a convex body in the Euclidean plane. In the first theorem the authors show that \[ 2\mu_1 (L)/\Delta (K) + \lambda_1(L) /D(K) \geq 1 \] and that the value 1 is attained for a triangle in the part of \(K\). The symbols \(2\mu_1(L)\) and \(\lambda_1(L)\) denote the maximal distance of two adjacent lattice lines, and the length of the shortest non-zero vector of \(L\), respectively. The symbols \(\Delta(K)\) and \(D(K)\) mean the minimal width and the diameter of \(K\). This inequality is a natural generalization of the inequality of \textit{P. R. Scott} [Bull. Lond. Math. Soc. 11, 273-278 (1979; Zbl 0413.52007)] who considered only integral lattices. The second theorem says that if \[ 2 \mu_1(L)/ \Delta(K) + \lambda_1(L)/D(K) \leq k^{-1} \] for a positive integer \(k\), then the interior of \(K\) contains at least \(k^2\) points of \(L\). The sign of weak inequality in the assumption of this theorem should be exchanged by sharp inequality; the present implication does not hold true for this triangle which gives equality in the first theorem.