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'$ such that $ |2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$ is defined to be the set $\left{a+b:a\neq b,\;a,b\in A\right}$. From this result, we get some lower bounds for $ |2\hspace{0.15cm}\widehat{} A| $. Finally, we give some remarks related to the problem for which sets $A\subset \mathbb{Z}{p}$ we have the equality $|2\hspace{0.15cm}\widehat{} A|=2|A|-1$.