# Unique Differences in Symmetric Subsets of $\mathbb{F}_p$

## TL;DR

This paper proves that for certain symmetric subsets of a finite field, there always exists an element with a unique difference representation, revealing new structural properties of these sets.

## Contribution

It establishes the existence of a uniquely representable difference in symmetric subsets of finite fields under specific size constraints, a novel result in additive combinatorics.

## Key findings

- Existence of a uniquely representable difference in symmetric subsets
- Condition on subset size relative to the field prime
- Structural insight into symmetric subsets of finite fields

## Abstract

Let $p$ be a prime and let $A$ be a subset of $\mathbb{F}_p$ with $A=-A$ and $|A\setminus\{0\}| \leq 2\log_3(p)$. Then there is an element of $\mathbb{F}_p$ which has a unique representation as a difference of two elements of $A$.