Problem Statement
Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,\[\lvert \# \{ n<m\leq n+k : x_m\in I\} - \lvert I\rvert k\rvert < \epsilon k.\]Is it true that, for every $\alpha$, the sequence $\{ \alpha p_n\}$ is not well-distributed, if $p_n$ is the sequence of primes?
Categories:
Analysis Discrepancy Primes
Progress
The notion of a well-distributed sequence was introduced by Hlawka and Petersen [Hl55].Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence $\{ \alpha n_k\}$ is not well-distributed for almost all $\alpha$.
He also claimed in [Er64b] to have proved that there exists an irrational $\alpha$ for which $\{\alpha p_n\}$ is not well-distributed. He later retracted this claim in [Er85e], saying "The theorem is no doubt correct and perhaps will not be difficult to prove but I never was able to reconstruct my 'proof' which perhaps never existed ."
The existence of such an $\alpha$ was established by Champagne, Le, Liu, and Wooley [CLLW24].
Source: erdosproblems.com/997 | Last verified: January 19, 2026