Fifty years of the Erd\H{o}s similarity conjecture

TL;DR

This paper reviews 50 years of progress on the Erdős similarity conjecture, focusing on new variants such as bi-Lipschitz, topological, and large-scale versions, highlighting recent research developments.

Contribution

It introduces and discusses recent advances on variants of the Erdős similarity conjecture, providing new perspectives and summarizing progress over five decades.

Findings

Analysis of bi-Lipschitz variant offers new insights.

Topological variant reveals different structural properties.

Large-scale variant extends the conjecture's scope.

Abstract

Erd\H{o}s similarity conjecture was proposed by P. Erd\H{o}s in 1974. The conjecture remains open for exponentially decaying sequences as well as Cantor sets that have both Newhouse thickness and Hausdorff dimension zero. In this article, written after 50 years of the conjecture being proposed, we review progress on some new variants of the original problem: namely, the bi-Lipschitz variant, the topological variant, and a variant ``in the large''. These problems were recently studied by the authors and their collaborators. Each of them offers new perspectives on the original conjecture.