Additive Combinatorics Using Equivariant Cohomology

TL;DR

This paper introduces a geometric approach using equivariant cohomology to address additive combinatorial problems, providing new proofs, improvements, and generalizations of classical theorems.

Contribution

It presents a novel geometric method employing equivariant cohomology to reprove, improve, and generalize key results in additive combinatorics.

Findings

Reproved the Dias da Silva-Hamidoune theorem

Improved a result of Sun on the Erdős-Heilbronn conjecture

Generalized G. Kós's Grashopper problem

Abstract

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erd\H{o}s-Heilbronn conjecture. We generalize a theorem of G. K\'os (the Grashopper problem) which in some sense is a simultaneous generalization of the Erd\H{o}s-Heilbronn conjecture. We also prove a signed version of the Erd\H{o}s-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.