AI 中文总结
该研究解决了格点公共铺砌函数支撑直径的公开差距问题，构造出满足条件的格点族，得到最优支撑直径上界并回答了相关问题。
AI 中文摘要
对于欧几里得空间R^d中体积为1且两两交集平凡的N个格点，每个非零公共铺砌函数的支撑直径为Ω(N^{1/d})；而对于基本域直径一致有界的格点族，标准卷积构造给出O(N)的上界，该差距自Kolountzakis和Wolff[1999年Mathematika的工作]以来一直是公开问题。我们解决了该差距，对所有d≥2及足够大的N，构造满足相同体积和交集条件的格点族，其容许非负公共铺砌函数，支撑直径为O(N^{1/d})，从而也回答了Kolountzakis和Papageorgiou[2022年jfaa的论文]的问题1。我们还得到最优O(√N)上界，对任意指定的平面格点族，其体积属于与N无关的固定有界集，构造出体积对应、两两交集平凡且基任意接近指定格点合适基的格点族。
英文摘要
For $N$ lattices in $\R^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $Ω(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff \cite{kolwolff-1999Mathematika}. We close this gap by constructing, for every $d\geq 2$ and all sufficiently large $N$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering Question 1 of Kolountzakis and Papageorgiou \cite{kolPapageorgiou-functions-2022jfaa}. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.
