Linear Algebraic Method and the Erd\H{o}s-Heilbronn Conjecture

TL;DR

This paper introduces a linear algebraic approach to prove the Erd ext{"o}s-Heilbronn conjecture, offering an elementary alternative to polynomial methods in additive combinatorics.

Contribution

It develops Das's linear algebraic method and provides a new elementary proof of the Alon-Nathanson-Ruzsa theorem, confirming the Erd ext{"o}s-Heilbronn conjecture.

Findings

Proves the Erd ext{"o}s-Heilbronn conjecture using linear algebra.

Provides an elementary proof avoiding polynomial methods.

Demonstrates the effectiveness of Vandermonde matrices in additive combinatorics.

Abstract

Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erd\H{o}s-Heilbronn conjecture. In this note, we develop Das's linear algebraic method and give a new elementary proof of the Alon-Nathanson-Ruzsa theorem for restricted sumsets, which implies the Erd\H{o}s-Heilbronn conjecture. Compared with the classical polynomial method via Combinatorial Nullstellensatz, our proof uses only basic linear algebra over finite fields, including Vandermonde matrices and solvability of linear systems.