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

2025/11/20 by Benjamin Bedert, Bedert, Benjamin
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.16636

openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

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 (1)/(n) from all other runners. A weaker lower bound of (1)/(2n-2) follows from the so-called trivial union bound, and subsequent work upgraded this to bounds of the form (1)/(2n)+(c)/(n²) for various constants c>0. Tao strengthened this to (1)/(2n)+\frac(log n)^1-o(1)n². In this paper, we obtain a polynomial improvement of the form (1)/(2n)+\frac1n^5/3+o(1).

Citations

• The lonely runner conjecture holds for eight runners
• Large sum-free subsets of sets of integers via L¹-estimates for trigonometric series
• Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants
• The Lonely Runner Conjecture turns 60
• Barely lonely runners and very lonely runners
• Some remarks on the lonely runner conjecture
• Correlation among runners and some results on the Lonely Runner\n Conjecture
• Random runners are very lonely
• The lonely runner with seven runners

Related

• Some remarks on the lonely runner conjecture
• On the time for a runner to get lonely
• Nine and ten lonely runners
• Lower bounds for the maximum number of runners that cause loneliness,\n and its application to Isolation
• Distribution of boundary points of expansion and application to the\n lonely runner conjecture