Higher Moments and Phase-Riesz Products for the Lonely Runner Conjecture
Year: 2026
Type: preprint
Abstract: For a set \(V\subset\mathbb N\) of distinct positive integers, write\[ \operatorname{ML}(V)=\max_{t\in\mathbb T}\min_{v\in V}\|tv\|_{\mathbb T}.\]The lonely runner conjecture of Wills and Cusick asserts that $\ML(V)\geqslant \frac{1}{n+1}$ for any such $V$ of size \(n\). In earlier work, the first author proved the uniform asymptotic lower bound\[ \operatorname{ML}(V)\ge \frac1{2n}+\frac1{n^{5/3+o(1)}}.\]In this note we refine the large-additive-dimension branch of that argument using a modified Riesz product construction, which yields the somewhat stronger bound\[ \operatorname{ML}(V)\ge \frac1{2n}+\frac1{n^{13/8+o(1)}}.\]The improvement to this exponent was obtained largely autonomously by ChatGPT 5.6 Pro.
Authors: Hu, Xiyu, BEDERT, BENJAMIN
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168571032
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