On the lattice point covering problem in dimension 2 (Q1740368)

Let $\mathcal{K}^{n}$ be the set of all convex bodies, i.e., compact convex sets in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$ with non-empty interior. We denote by $\mathcal{K}^{n}_{(\circ)} \subset \mathcal{K}^{n}$ the subset of all convex bodies having the origin as an interior point, i.e., $0 \in \text{int} (K)$, and by $\mathcal{K}^{n}_{(s)} \subset \mathcal{K}^{n}_{(\circ)}$ the set of those bodies which are symmetric with respect to $0$. We say that a convex body $K \in \mathcal{K}^{n}$ has the lattice point covering property if $K$ contains a lattice point of $\mathbb{Z}^{n}$ in any position, i.e., in any translation and rotation of $K$. Let $K \in \mathcal{K}^{n}$, we denote by $Z(K)$ the lattice point covering radius of $K$, i.e., the smallest positive number $r$ such that $rK$ has the lattice point covering property. In the paper under review, the author studies the lattice point covering properties of regular $n$-gons $H_{n}$ in $\mathbb{R}^{2}$. \par Main Theorem. Let $t_{i} > 0$ and $i \in \mathbb{N}$. The following conditions are equivalent. \begin{itemize} \item[i)] $t_{4n} \cdot H_{4n}$ contains a lattice point of $\mathbb{Z}^{2}$ in any position; \item[ii)] $t_{4n} \cdot H_{4n}$ contains a ball with radius $\frac{\sqrt{2}}{2}$; \item[iii)] $t_{4n} \geq \frac{ \frac{\sqrt{2}}{2}}{\cos (\frac{\pi}{2n})}$. \end{itemize} If $n=1, 2$, then the following conditions are equivalent. \begin{itemize} \item[i)] $t_{4n+2} \cdot H_{4n+2}$ contains a lattice point of $\mathbb{Z}^{2}$ in any position; \item[ii)] $t_{4n+2} \cdot H_{4n+2}$ contains $\left[-\frac{1}{2}, \frac{1}{2} \right]^{2}$; \item[iii)] $t_{6} \geq \frac{1}{3-\sqrt{3}}$, $t_{10} \geq \frac{\cos(\pi/5) - \sin (\pi/5) + \sin(2\pi/5) - \cos(2\pi/5)}{2 \sin(\pi/5)}$. \end{itemize}