On unique sums in Abelian groups

Let $A$ be a subset of the cyclic group $\mathbf{Z}\mathrm{/}p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $\mathrm{\mid }A\mathrm{\mid }$ 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 $a_1^{\mathrm{\prime }},a_2^{\mathrm{\prime }}\in A$ such that $a_1+a_2=a_1^{\mathrm{\prime }}+a_2^{\mathrm{\prime }}$ and $\{a_1,a_2\}\mathrm{\ne }\{a_1^{\mathrm{\prime }},a_2^{\mathrm{\prime }}\}$. Let $m(p)$ be the size of a smallest subset of $\mathbf{Z}\mathrm{/}p\mathbf{Z}$ 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 $\omega (p)\log p⩽m(p)\ll (\log p{)}^2$ for some function $\omega (p)$ which tends to infinity as $p\to \mathrm{\infty }$. In particular, this shows that for any $B\subset \mathbf{Z}\mathrm{/}p\mathbf{Z}$ of size $\mathrm{\mid }B\mathrm{\mid }<\omega (p)\log p$, 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.