Problem Statement
Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?
Categories:
Combinatorics
Progress
The Erdős similarity problem.This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.
Steinhaus [St20] has proved this is false whenever $A$ is a finite set.
This conjecture is known in many special cases (but, for example, it is open when $A=\{1,1/2,1/4,\ldots\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen [JLM24].
Source: erdosproblems.com/120 | Last verified: January 13, 2026