Geometric Ruzsa triangle inequality in metric spaces with dilations

TL;DR

This paper extends the geometric Ruzsa triangle inequality to metric spaces with dilations, broadening its applicability beyond algebraic or incidence structures, and providing a new tool for geometric analysis.

Contribution

It introduces a novel version of the Ruzsa triangle inequality applicable to metric spaces with dilations, without relying on algebraic or incidence structures.

Findings

Established a geometric Ruzsa triangle inequality in metric spaces with dilations

Demonstrated the inequality's applicability without algebraic structures

Extended the inequality's use to broader geometric contexts

Abstract

The Appendix of the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev, contains a general "geometric Ruzsa triangle inequality" in a Desarguesian projective plane. The purpose of this note is to give a similar inequality for metric spaces with dilations, that is in the absence of an algebraic or incidence structure.