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.

设A是循环群\({\textbf{Z}}/p{\textbf{Z}}\)的子集，其中p为素数。确定| 有多小是一个经过充分研究的问题。一个|如果\(A+A\)中没有唯一的和，则可以是，这意味着对于A\ 中的每两个元素 \(a_1,a_2\)，存在A\(a_1',a_2'\in A\)使得\(a_1+a_2=a_1'+a_2'\)和\(\{a_1,a_2\}\ne \{a_1',a_2'\}\)。令m ( p )为没有唯一和的最小子集\({\textbf{Z}}/p{\textbf{Z}}\)的大小。之前最知名的边界是\(\log p \ll m(p)\ll \sqrt{p}\)。在本文中，我们将某些函数\(\omega (p)\) 的上限和下限改进为\(\omega (p)\log p \leqslant m(p)\ll (\log p)^2\) )趋于无穷大为\(p\rightarrow \infty \)。特别是，这表明对于任何大小为\(|B|<\omega (p)\log p\)的 \ (B\subset {\textbf{Z}}/p{\textbf{Z}}\)，它的和集\(B+B\)包含唯一的和。我们还获得了没有唯一和的一般阿贝尔群的最小子集的大小的相应界限。