Asymptotically tight bounds on subset sums

For a subset A of a finite abelian group G we define Σ(A) = {∑a∈B a : B ⊂ A}. In the case that Σ(A) has trivial stabiliser, one may deduce that the size of Σ(A) is at least quadratic in |A|; the bound |Σ(A)| ≥ |A|2/64 has recently been obtained by De Vos, Goddyn, Mohar and Šámal [2]. We improve this bound to the asymptotically best possible result |Σ(A)| ≥ (1/4− o(1))|A|2. We also study a related problem in which A is any subset of Zn with all elements of A coprime to n; it has recently been shown, by Vu [6], that if such a set A has the property Σ(A) 6= Zn then |A| = O(√n). This bound was improved to |A| ≤ 8√n by De Vos, Goddyn, Mohar and Šámal [2], we further improve the bound to the asymptotically best possible result |A| ≤ (2 + o(1))√n.