发表机构
University of Otago（奥塔哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过构造ℝ⁵⁶中两两ℓ₅距离相等的58个点，否定了Kusner关于2<p<∞时ℓₚⁿ中等边集最多含n+1个点的猜想，确定猜想不成立的指数下确界在[4,5)区间。
AI 中文摘要
我们否定了Kusner在1983年提出的猜想：对于2<p<∞的ℓₚⁿ空间，其中的每个等边集最多有n+1个点；具体而言，在ℝ⁵⁶中存在58个点，它们两两之间的ℓ₅距离均相等，因此最大等边集大小满足e(ℓ₅⁵⁶)≥58>57。这是有限p≥2时，ℓₚⁿ空间中首个点数超过n+1的等边集。该构造在指数5附近的开区间上成立；由于Ge、Xu和Zhou近期证明了2≤p≤4时该猜想成立，因此猜想不成立的指数下确界位于[4,5)区间内。该构型是有理盒中具有有理系数的显式多项式方程组的唯一解，通过精确算术建立。
英文摘要
We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2<p<\infty$ has at most $n+1$ points: there exist $58$ points in $\mathbb{R}^{56}$ whose pairwise $\ell_5$ distances are all equal, so the maximum equilateral-set size satisfies $e(\ell_5^{56})\ge58>57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.
CommentsCertificate data and exact verifier: https://doi.org/10.5281/zenodo.21911503, I have amended this manuscript over a previous version to reduce its reliance on supplemental material. I also specify the search method
