Cauchy-Davenport Theorem for linear maps: Simplification and Extension

TL;DR

This paper provides a new combinatorial proof of the Cauchy-Davenport Theorem for linear maps, extending its applicability and offering insights into previously unaddressed parameter ranges.

Contribution

It introduces a purely combinatorial proof of the theorem and extends its scope to new parameter ranges not covered before.

Findings

New combinatorial proof of the Cauchy-Davenport Theorem for linear maps

Extension of the theorem to additional parameter ranges

Insights into the behavior of linear maps on grids

Abstract

We give a new proof of the Cauchy-Davenport Theorem for linear maps given by Herdade et al., (2015). This theorem gives a lower bound on the size of the image of a linear map on a grid. Our proof is purely combinatorial and offers a partial insight into the range of parameters not handled previously.