Linear Algebraic Method and the Erdős-Heilbronn Conjecture

Abstract

Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erdős-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 Erdős-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.