Sets with few subset sums

Abstract: A classical result of Nathanson shows 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. We establish stability versions of this inverse theorem in two regimes. First, for any parameter $0 \leq M \leq n-4$, 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. Joint work with Ruben Carpenter and Colin Defant.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)