Mixed thresholds in the Lonely Runner Conjecture
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| Format: | Preprint |
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2026
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| _version_ | 1866918526585929728 |
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| author | Jensen, Alathea |
| author_facet | Jensen, Alathea |
| contents | The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$.
We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$.
We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_27941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mixed thresholds in the Lonely Runner Conjecture Jensen, Alathea Number Theory Combinatorics 11J13, 05D99 G.2.1 The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$. We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$. We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds. |
| title | Mixed thresholds in the Lonely Runner Conjecture |
| topic | Number Theory Combinatorics 11J13, 05D99 G.2.1 |
| url | https://arxiv.org/abs/2605.27941 |