On restricted sumsets over a field

Abstract

We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1⩽i\ne j⩽n)$ are finite subsets of $F$ with cardinality $m$, and $A_1,\dots ,A_n$ are finite nonempty subsets of $F$ with $|A_1|=\cdots =|A_n|=k$. Let $p(F)$ be the additive order of the identity of $F$. It is proved that $|C|⩾\min \{p(F),n(k-1)-mn(n-1)+1\}$ if $p(F)>mn$. This conclusion refines the result of Hou and Sun.

accepted by Finite Fields and Their Applications