Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness

TL;DR

This paper improves the lower bound on the minimum separation distance in the lonely runner conjecture using Riesz products, achieving a polynomial enhancement over previous bounds.

Contribution

It introduces a novel approach employing Riesz products to significantly improve the known bounds for the lonely runner conjecture.

Findings

Achieved a polynomial improvement of the lower bound to 1/(2n) + 1/n^{5/3+o(1)}.

Demonstrated the effectiveness of Riesz products in analyzing the lonely runner problem.

Extended the theoretical understanding of separation bounds in the conjecture.

Abstract

The lonely runner conjecture of Wills and Cusick asserts that if $n$ runners with distinct constant speeds run around a a circular unit length track, starting at a common time and place, then each runner will at some time be separated by a distance of at least $\frac{1}{n}$ from all other runners. A weaker lower bound of $\frac{1}{2n-2}$ follows from the so-called trivial union bound, and subsequent work upgraded this to bounds of the form $\frac{1}{2n}+\frac{c}{n^2}$ for various constants $c>0$. Tao strengthened this to $\frac{1}{2n}+\frac{(\log n)^{1-o(1)}}{n^2}$. In this paper, we obtain a polynomial improvement of the form $$\frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}.$$