← 研踪 PaperTrailarXiv:2611.16377 · v1
Universal optimality on finite Lee tori
首次提交 2026-11-12 14:44
版本更新 2026-11-12 14:44
雷达观测 2026-11-13 14:15
摘要
We study subsets of finite Lee tori that minimize pair energy among sets of the same cardinality for every discretely completely monotone potential. At half occupancy on a product of even cycles, the two checkerboards minimize both total pair energy and maximum individual energy. They are the only minimizers for every nonconstant classical completely monotone potential on $(0,\infty)$, proving the checkerboard conjecture of Bouman, Draisma, and van Leeuwaarden for this class. At the independence number of the distance one graph on $\Z_q^n$, universally optimal sets exist exactly when $q$ is even, $q=3$, $n=1$, or $(q,n)=(5,2)$. In these cases they are precisely the maximum independent sets and form a single Lee isometry class. At density $1/5$ in $\Z_5^n$, $n\ge2$, universal optimality is characterized by a weighted count of pairs at distances one and two. We prove these results using integer box occupancy bounds that strengthen Fourier linear programming and, in the quinary case, Fourier certificates with a common zero set.
作者备注：30 pages
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v1 · 2026-11-12 14:44 · 最新观测版本
Universal optimality on finite Lee tori
We study subsets of finite Lee tori that minimize pair energy among sets of the same cardinality for every discretely completely monotone potential. At half occupancy on a product of even cycles, the two checkerboards minimize both total pair energy and maximum individual energy. They are the only minimizers for every nonconstant classical completely monotone potential on $(0,\infty)$, proving the checkerboard conjecture of Bouman, Draisma, and van Leeuwaarden for this class. At the independence number of the distance one graph on $\Z_q^n$, universally optimal sets exist exactly when $q$ is even, $q=3$, $n=1$, or $(q,n)=(5,2)$. In these cases they are precisely the maximum independent sets and form a single Lee isometry class. At density $1/5$ in $\Z_5^n$, $n\ge2$, universal optimality is characterized by a weighted count of pairs at distances one and two. We prove these results using integer box occupancy bounds that strengthen Fourier linear programming and, in the quinary case, Fourier certificates with a common zero set.
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