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

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 $$\frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}.$$