发表机构
The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该数学研究针对n≥5时子集和较少集合的例外边界，确定了M=n-3和M=n-2两种情况的集合分类，推导过渡公式并证明n=5为严格阈值。
AI 中文摘要
设A为n个正实数构成的集合，记FS(A)为其所有子集和（包含空集的和），记T_n=binom(n+1,2)。对于0≤M≤n-4，Carpenter、Defant和Kravitz对所有满足|FS(A)|≤T_n+1+M的集合A进行了分类。我们确定了n≥5时的接下来两种情况：当M=n-3时，这类集合恰好是原初正整数集合B的正伸缩，且满足∑B≤T_n+n-3，以及{1,3,4,…,n+1}的伸缩；当M=n-2时，整数总和边界增加1，例外模板为{1,3,4,…,n+1}、{2,3,…,n+1}和{1,3,4,…,n,n+2}。我们还确定了所有等号情况，证明n=5是严格阈值，包括M=n-2时的可公度性，证明过程记录了两个端点处缺失的子集和，并推导了一个过渡公式，将例外构型推广到任意n。
英文摘要
Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\binom{n+1}{2}$. For every fixed integer $C\geq-1$ and all sufficiently large $n$, we classify the sets satisfying $$ |FS(A)|\leq T_n+n+C+1. $$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\sum B\leq T_n+n+C$ or belongs to an explicit exceptional family specified by a missing element $m\in\{1,2\}$ and an integer partition of $C+m$ or $C+m+1$. If $P$ denotes the partition function, the exceptional family has exactly $$ P(C+1)+2P(C+2)+P(C+3) $$ primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large $i$, adjoining the largest element to the preceding $i-1$ elements creates only $i+e$ new subset sums, where $e$ is bounded, then the $i$-element set is a dilation of $[1,i+e]_{\mathbb{Z}}$ with exactly $e$ elements deleted. Conversely, every such deletion pattern has increment excess $e$. The proof combines a stabiliser argument in $\mathbb{R}/x\mathbb{Z}$, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for $C\leq-2$, this completes the eventual fixed-defect classification for every integer $C$.
Comments22 pages, no figures. This version is a sequel establishing a bounded-excess inverse theorem and an eventual fixed-defect classification for subset sums. It builds on the sharp C=-3 and C=-2 boundary classifications and endpoint results in arXiv:2609.00044v2
