DISPROVED
This has been solved in the negative.
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite such that $a_{i+1}/a_i\to 1$. For any $x\geq a_1$ let\[f(x) = \frac{x-a_i}{a_{i+1}-a_i}\in [0,1),\]where $x\in [a_i,a_{i+1})$. Is it true that, for almost all $\alpha$, the sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$?
For example if $A=\mathbb{N}$ then $f(x)=\{x\}$ is the usual fractional part operator.
A problem due to Le Veque
[LV53], who proved it in some special cases. Davenport and LeVeque
[DaLe63] proved this under the assumption that $a_n-a_{n-1}$ is monotonic. Davenport and Erdős
[DaEr63] proved it is true if $a_n \gg n^{1/2+\epsilon}$ for some $\epsilon>0$.
The general conjecture is false, as shown by Schmidt
[Sc69].
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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 #492, https://www.erdosproblems.com/492, accessed 2026-01-16
