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| Hovedforfatter: | |
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| Format: | Preprint |
| Udgivet: |
2019
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| Fag: | |
| Online adgang: | https://arxiv.org/abs/1908.02153 |
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Indholdsfortegnelse:
- In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have $$ ||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)π}{k-1} $$ for all $i=1,\ldots,k-1$ and where $1>\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running around a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound $$ ||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{Cπ}{7} $$ for some constant $1>C>0$ for all $1\leq i\leq 7$.