The Erd\H{o}s 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 \cite{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 \cite{Sv00} on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen \cite{JLM24}.


References


[JLM24] Y. Jung and C.-K. Lai and Y. Mooroogen, Some recent progress on the Erd\H{o}s similarity conjecture. arXiv:2412.11062 (2024).

[St20] Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.

[Sv00] Svetic, R. E., The Erd\H{o}s similarity problem: a survey. Real Anal. Exchange (2000/01), 525-539.