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

TL;DR

This paper advances inverse theorems in additive combinatorics by proving that for sets in  with specific sumset sizes, the sets must be equal and form an arithmetic progression, refining previous results.

Contribution

It establishes that when the restricted sumset size reaches a certain threshold, the sets are necessarily equal and structured as arithmetic progressions.

Findings

If  has more than 2k-2 elements, the restricted sumset size is at least 2k-3.

Equality in sumset size implies the sets are equal and form an arithmetic progression.

The result extends previous inverse theorems for restricted sumsets in finite fields.

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.