The Erdős similarity problem: a survey
A~set~\(E\) of reals is said to be Erdős if there exists a set of positive Lebesgue measure not containing a copy of \(E\), i.e., a set of the form \(x+tE=\{x+te:e\in E\}\) with \(x,t\in\mathbb{R}\), \(t\neq 0\). The Erdős similarity problem asks whether every infinite set is Erdős. The problem can be easily reduced to zero-sequences but the proof is known for only a few special cases not including the geometric sequence \(\{2^{-n}\}_{n=1}^\infty\). The author surveys the known results on Erdős sets and related properties, presents some new results and asks several related questions.
scientific article; zbMATH DE number 1848857