SOLVED

Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that there are infinitely many $m,n$ such that
\[\lvert x_{m+n}-x_n\rvert \leq \frac{1}{\sqrt{5}n}?\]

A conjecture of Newman. This was proved Chung and Graham, who in fact show that for any $\epsilon>0$ there must exist some $n$ such that there are infinitely many $m$ for which
\[\lvert x_{m+n}-x_m\rvert < \frac{1}{(c-\epsilon)n}\]
where
\[c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.535\cdots\]
and $F_m$ is the $m$th Fibonacci number. This constant is best possible.