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

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.