Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness
Benjamin Bedert
TL;DR
The paper investigates the Lonely Runner Conjecture by establishing a polynomial improvement for the maximal loneliness $ML(V)$: $ML(V)\ge \frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}$. The authors split the analysis into two additive-dimension regimes: for large $\dim_2(V)$ they construct a Riesz-product witness to push the lower bound beyond the trivial $1/(2n)$ by a factor proportional to $\dim_2(V)^2$, while for small $\dim_2(V)$ they transfer to a dense model $V'$ with a sharp bound on $V'$ supported in a short interval via a mod-$p$ averaging argument. Intermediate steps include a dissociated-set Loneliness bound and model-transfer lemmas that may be of independent interest, together with a density-based refinement that recovers and strengthens prior log-improvements. The work advances the structure-versus-randomness understanding of the problem and offers a new harmonic-analytic toolkit (Riesz products, Bohr-sets, and 2-dissociation) to push the Lonely Runner gap toward the conjectured threshold.
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 $$\frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}.$$