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Erdős Problem #480
PROVED
This has been solved in the affirmative.
Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that\[\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447?\]
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).
This page was last edited 28 December 2025.
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Additional thanks to: Salvatore Mercuri and Wouter van Doorn
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #480, https://www.erdosproblems.com/480, accessed 2025-12-31
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #480, https://www.erdosproblems.com/480, accessed 2025-12-31