发表机构
University of Toronto Scarborough（多伦多大学士嘉堡分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限实数集的最小递增整数模型，在保持等长和相等关系下，确定了$H_4(q)$、$H_5(q)$和$H_6(2)$的精确值，并给出更大集合的渐近界及应用。
AI 中文摘要
我们研究一个有限实数集的最小递增整数模型可以有多小，同时保持所有长度至多$q$、允许重复的元素等长和之间的相等关系。递增对应必须精确保持哪些和相等；不等比较的符号可以改变。设$H_m(q)$为足以容纳每个大小为$m$的有序实数集的最小直径。对于每个整数$q\ge2$，我们证明$H_4(q)=q(q+1)$和$H_5(q)=q^2(q+1)$，并确定$H_6(2)=24$。证明使用了由加法等式和规定元素顺序的不等式所确定的线性空间。这些证明还给出了更大集合的界，确定了每个固定$m\ge4$的前两个渐近项，并产生了进一步的锐利示例族。一个应用限制了实现有限加法无平方谱所需的整数字母表。补充数据支持有限的六元素分类。
英文摘要
We ask how small an increasing integer model of a finite real set can be while preserving all equalities between equal-length sums of at most $q$ elements, with repetitions allowed. The increasing correspondence must preserve exactly which sums are equal; the signs of unequal comparisons may change. Let $H_m(q)$ be the least diameter sufficient for every ordered real set of size $m$. For every integer $q\ge2$ we prove $H_4(q)=q(q+1)$ and $H_5(q)=q^2(q+1)$, and we determine $H_6(2)=24$. The proofs use the linear space determined by the additive equalities and the inequalities prescribing the order of the elements. They also give bounds for larger sets, determine the first two asymptotic terms for each fixed $m\ge4$, and yield further families of sharp examples. An application bounds the integer alphabets needed to realize finite additive-square-free spectra. The supplementary data support the finite six-element classification.
Comments31 pages, including the technical supplement. Computational data are available at https://doi.org/10.5281/zenodo.22563136
