Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912721412292608 |
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| author | Bedert, Benjamin |
| author_facet | Bedert, Benjamin |
| contents | 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 $$\frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}.$$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness Bedert, Benjamin Combinatorics Number Theory 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 $$\frac{1}{2n}+\frac{1}{n^{5/3+o(1)}}.$$ |
| title | Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2511.16636 |