On Unique Sums in Abelian Groups

Let A be a subset of the cyclic group \({\textbf{Z}}/p{\textbf{Z}}\) with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in \(A+A\), meaning that for every two elements \(a_1,a_2\in A\), there exist \(a_1',a_2'\in A\) such that \(a_1+a_2=a_1'+a_2'\) and \(\{a_1,a_2\}\ne \{a_1',a_2'\}\). Let m(p) be the size of a smallest subset of \({\textbf{Z}}/p{\textbf{Z}}\) with no unique sum. The previous best known bounds are \(\log p \ll m(p)\ll \sqrt{p}\). In this paper we improve both the upper and lower bounds to \(\omega (p)\log p \leqslant m(p)\ll (\log p)^2\) for some function \(\omega (p)\) which tends to infinity as \(p\rightarrow \infty \). In particular, this shows that for any \(B\subset {\textbf{Z}}/p{\textbf{Z}}\) of size \(|B|<\omega (p)\log p\), 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.