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

TL;DR

This paper introduces a transformation technique for subsets of rac{p}{Z} that helps establish lower bounds on the size of restricted sumsets, and discusses conditions for equality cases.

Contribution

It presents a procedure to transform sets in rac{p}{Z} to analyze and bound their restricted sumset sizes, advancing understanding of sumset cardinalities.

Findings

A transformation procedure for sets in rac{p}{Z} that preserves sumset size bounds.

New lower bounds for the size of restricted sumsets in rac{p}{Z}.

Remarks on conditions for sumset size equality rac{p}{Z}.

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 $|2\hspace{0.15cm}\widehat{} A|=2|A|-1$.