Fixed-Defect Inverse Theorems for Subset Sums

Abstract

Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$. For every fixed integer $C\ge -1$ and all sufficiently large $n$, we classify the sets satisfying $$|FS(A)|\le T_n+n+C+1.$$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\sum B\le T_n+n+C$ 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 $C\le -2$, this completes the eventual fixed-defect classification for every integer $C$.

arXiv comment22 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