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 $T_n=\binom{n+1}{2}$. 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 $C$.