Coprime Mappings and Lonely Runners
| Autor: | Bohman, Tom, Peng, Fei |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number $t$ such that $ 1/(n+1) \le \{ v_i t\} \le n/(n+1) $ for $ i = 1, \dots, n$. In this paper we prove that if $ \epsilon >0 $ and $n$ is sufficiently large (relative to $\epsilon$) then such a $t$ exists for any collection of positive integers $ v_1 < v_2 < \dots < v_n$ such that $ v_n < (2-\epsilon)n$. This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao. The key ingredient in our proof is a result on coprime mappings. Let $A$ and $B$ be sets of integers. A bijection $ f:A \to B$ is a coprime mapping if $ a $ and $f(a)$ are coprime for every $ a \in A$. We show that if $A,B \subset [n]$ are intervals of length $2m$ where $ m = e^{ \Omega({(\log\log n)}^2)}$ then there exists a coprime mapping from $A$ to $B$. We do not believe that this result is sharp. Comment: 26 pages, 1 figure |
| Databáze: | arXiv |
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