Problem 997

Erdős Problem #997: declared status 'proved'. Formalized: yes. 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? Prize: no. Tags: analysis, discrepancy, primes.