Asymptotically tight bounds on subset sums

TL;DR

This paper establishes the tight asymptotic lower bounds on the size of subset sums in finite abelian groups and related sets in Z_n, improving previous bounds to their optimal asymptotic forms.

Contribution

The paper proves the asymptotically optimal lower bound |Sigma(A)| >= (1/4 - o(1))|A|^2 for subset sums with trivial stabilizer and improves bounds on the size of sets in Z_n with non-complete subset sums.

Findings

|Sigma(A)| >= (1/4 - o(1))|A|^2 for subsets with trivial stabilizer

|A| <= (2 + o(1))sqrt{n} for subsets in Z_n with non-complete Sigma(A)

Improved bounds are asymptotically tight and optimal

Abstract

For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.