Sets with Few Subset Sums

TL;DR

The paper investigates the structure of n-element sets of positive reals with few subset sums, providing precise characterizations and stability results for various bounds, extending classical combinatorial theorems.

Contribution

It offers new stability theorems and characterizations for sets with few subset sums, refining classical results and exploring higher-dimensional analogs.

Findings

Characterized sets with at most inom{n+1}{2}+1+M subset sums for M extless n-4.

Provided a sharp characterization of sets with at most Cn^2 subset sums.

Constrained the structure of n-element subsets in ext{R}^d with o(n^{d+1}) subset sums.

Abstract

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of this inverse theorem in two regimes. First, for any parameter $M \leq n-4$, we precisely characterize the $n$-element sets of positive reals with at most $\binom{n+1}{2}+1+M$ subset sums. Second, for any constant $C$, we provide a characterization, sharp up to constants, of the $n$-element sets of positive reals with at most $Cn^2$ distinct subset sums. Along the way, we constrain (for any fixed $d \geq 2$) the structure of $n$-element subsets of $\mathbb{R}^d$ with $o(n^{d+1})$ subset sums.