On Unique Sums in Abelian Groups

Let A be a subset of the cyclic group Z/pZ with p prime. It is a well-studied problem to determine how small vertical bar A vertical bar can be if there is no unique sum in A+A, meaning that for every two elements a(1),a(2) is an element of A, there exist a(1)',a(2) 'is an element of A such that a(1) + a(2) = a(1)'+a(2)' and {a(1), a(2)'} not equal {a(1)', a(2)'}. Let m(p) be the size of a smallest subset of Z/pZ with no unique sum. The previous best known bounds are log p << m (m) << root p. In this paper we improve both the upper and lower bounds to omega(p)log p <= m(p) << (log P)(2) for some function omega(P) which tends to infinity as P -> infinity. In particular, this shows that for any B subset of Z/(P)Z of size vertical bar B vertical bar < omega(P) log P, its sumset B + B contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.