Bounds on the cardinality of restricted sumsets in ${\mathbb{Z}}_p$

Abstract

In this paper we present a procedure which allows to transform a subset $A$ of ${\mathbb{Z}}_p$ into a set $A^{\prime }$ such that $|2\hat{}{A}^{\prime }|\le |2\hat{}A|$, where $2\hat{}A$ is defined to be the set $\left\{a+b:a\ne b,a,b\in A\right\}$. From this result, we get some lower bounds for $|2\hat{}A|$. Finally, we give some remarks related to the problem for which sets $A\subset {\mathbb{Z}}_p$ we have the equality $|2\hat{}A|=2|A|-1$.

arXiv comment16 pages