AI 中文总结
研究平面周期点集的硬币覆盖问题，通过构造周期图案，探讨三角格、方格和蜂窝点集可被两两不重叠单位圆盘覆盖的间距区间，修正了方格图案的重叠问题并给出新图案。
AI 中文摘要
我们研究平面上周期点集的硬币覆盖问题的无限变体。给定间距为\(d\)的点集，询问其所有点是否能被两两不重叠的单位圆盘覆盖。我们考虑三角格、方格和蜂窝点集，构造周期图案来证明可覆盖间距的若干区间。对于三角格，构造包括具有顶点、面和非格点实现中心的单族图案以及多族图案。对于蜂窝点集，额外的原生图案填补三角格构造留下的间隙。对于方格，重新审视Alm等人的构造，识别出一个图案实现中的意外重叠，并给出新图案来恢复部分受影响区间并建立一个额外的可覆盖区间。
英文摘要
We study an infinite variant of the coin-covering problem for periodic point sets in the plane. Given a point set of spacing $d$, we ask whether all of its points can be covered by pairwise non-overlapping unit disks. We consider the triangular lattice, the square lattice, and the honeycomb point set, and construct periodic motif patterns that certify several intervals of coverable spacings. For the triangular lattice, our constructions include single-family patterns with vertex, face, and off-lattice realizing centers, as well as multi-family patterns. For the honeycomb point set, additional native motifs fill gaps left by the triangular-lattice constructions. For the square lattice, we revisit the constructions of Alm et al., identify an unintended overlap in one motif realization, and give new patterns that recover part of the affected interval and establish an additional coverability interval.
CommentsSubmitted to JCDCGGG'26
