Fifty years of the Erdős similarity conjecture
Yeonwook Jung, Chun-Kit Lai, Yuveshen Mooroogen
TL;DR
This survey synthesizes five decades of progress on the Erdős similarity conjecture, focusing on measure universality, bi-Lipschitz variants, uncountable Cantor sets, topological universality, and a large-scale dual problem. It highlights key techniques—sumset reformulations, Newhouse thickness, Frostman and martingale methods, and metric number theory—and presents both obstacles and breakthroughs for sequences, fractal sets, and lacunary versus slow-decaying regimes. The work clarifies how rigidity or flexibility of embeddings (affine vs bi-Lipschitz) influences universality and connects deterministic combinatorial structure with probabilistic constructions and descriptive-set-theoretic insights. Collectively, the results delineate when large sets necessarily contain prescribed patterns and reveal deep connections between fractal geometry, additive combinatorics, and ergodic/density phenomena, while outlining substantial open questions and potential dichotomies. The findings have implications for understanding ubiquitous patterns in large sets and for the broader theory of pattern detection in fractal and metric-number-theoretic contexts.
Abstract
Erdős similarity conjecture was proposed by P. Erdős 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.