发表机构
University of Toronto Scarborough（多伦多大学士嘉堡校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究四元素整数集合的$h$重和集基数所需的最小区间直径，统一证明$h\ge17$时的精确公式，并通过有限搜索和构造给出完整刻画。
AI 中文摘要
对于四元素整数集合的$h$重和集，实现所有可达到基数所需的最小区间直径为：当$2\le h\le10$时，为$\binom{h+2}{2}+(h\bmod2)$；当$h\ge11$时，为$d_h=(h-3)(h-1)$。我们统一证明了$h\ge17$时的公式，并利用完全有限搜索以及关系秩覆盖论证处理其余阶数。对于下界，我们证明当$h\ge9$时，每个具有$\binom{h+3}{3}-17$个不同$h$重和的整数四元集合，其归一化直径属于$\{d_h,d_h+1,d_h+2\}$，并构造达到$d_h$的集合。对于上界，关系格指标论证将较大的最小直径代表限制为两个或三个活跃方向。两个一般构造和四个特殊缺陷随后提供了实现所有剩余基数的短集合。
英文摘要
The least interval diameter needed to realize every attainable cardinality of an $h$-fold sumset of a four-element integer set is $\binom{h+2}{2}+(h\bmod2)$ for $2\le h\le10$ and $d_h=(h-3)(h-1)$ for $h\ge11$. We prove the formula uniformly for $h\ge17$ and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with $\binom{h+3}{3}-17$ distinct $h$-fold sums has normalized diameter in $\{d_h,d_h+1,d_h+2\}$ when $h\ge9$, and we construct sets attaining $d_h$. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.
Comments19 pages, 4 tables. Code and exact computational data are provided as ancillary files
