On the Elementary Proof of the Inverse Erd\H{o}s-Heilbronn Problem

TL;DR

This paper provides an elementary proof for the inverse Erdős-Heilbronn problem in integers, avoiding advanced algebraic methods, and offers partial results for the problem in modular arithmetic.

Contribution

It introduces a fully elementary proof for the inverse Erdős-Heilbronn problem in  and partial results in modular settings, bypassing complex algebraic techniques.

Findings

Elementary proof established for  case

Partial results obtained for /p case

Avoids polynomial method and Nullstellensatz

Abstract

In this article, we studied the inverse Erd\H{o}s-Heilbronn problem with the restricted sumset from two components $A$ and $B$ that are not necessarily the same. We give a completely elementary proof for the problem in $\mathbb{Z}$ and some partial results that contributes to the elementary proof of the problem in $\mathbb{Z}/p\mathbb{Z}$, avoiding the usage of the powerful polynomial method and the Combinatorial Nullstellensatz.