Problem #480: Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence

---

Progress

A conjecture of Newman. This was proved by Chung and Graham [ChGr84], who in fact prove that\[\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq \frac{1}{c}\approx 0.3944\]where\[c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.535\cdots\]and $F_m$ is the $m$th Fibonacci number. They also prove that this constant is best possible (van Doorn discusses their construction in the comments).