OPEN
This is open, and cannot be resolved with a finite computation.
- $100
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 $a\neq 0$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
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].
View the LaTeX source
This page was last edited 23 January 2026. View history
External data from
the database - you can help update this
Formalised statement?
Yes
|
Likes this problem
|
None
|
|
Interested in collaborating
|
None
|
|
Currently working on this problem
|
None
|
|
This problem looks difficult
|
Vjeko_Kovac
|
|
This problem looks tractable
|
None
|
|
The results on this problem could be formalisable
|
None
|
|
I am working on formalising the results on this problem
|
None
|
Additional thanks to: Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #120, https://www.erdosproblems.com/120, accessed 2026-04-11
