Call $x_1,x_2,\dots \in (0,1)$ well-distributed if, for every $ϵ>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,$$|\#\{n<m\le n+k:x_m\in I\}-|I|k|<ϵk.$$Is it true that, for every $\alpha$, the sequence $\{\alpha p_n\}$ is not well-distributed, if $p_n$ is the sequence of primes?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Your definition is the standard “well-distributed (mod 1)” condition (uniform equidistribution **in every long block**, uniformly over the starting point). In the literature it is often stated as: for every interval ([a,b]\subset[0,1]), [ \lim_{N\to\infty}\ \sup_{m\in\mathbb N}\left|\frac{1}{N}|\\{1\le n\le N:\ a\le {s…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.