One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$

Abstract

Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$, $a\neq b$. The number of elements of this set is at least $2k-3$. K\'{a}rolyi showed that, except from some particular cases, The equality can only occur if $A = B$ and $A$ is an arithmetic progression with non zero difference. We prove that in the case that $|A\dot{+}B| = 2k - 2$ and $|A|=|B|$ the equality $A=B$ holds.