An old result of Nathanson shows that every n-element set of positive reals has at least n(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 M at most n-4, we precisely characterize the n-element sets of positive reals with at most n(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 subset sums.

Joint work with Ruben Carpenter and Colin Defant.