Lonely Runner Conjecture- II

by ayushkhaitan3437

The Lonely Runner conjecture states that each runner is lonely at some point in time. Let the speeds of the runners be \{a_1,a_2,\dots,a_k\}, and let us prove “loneliness” for the runner with speed a_e.

As we know, the distance between a_i and a_e is given by the formula |a_e-a_i|t for t\leq \frac{1}{2|a_e-a_i|} and 1-|a_e-a_i|t for t\geq \frac{1}{2|a_e-a_i|}, where t stands for time. Also, speed\times time=distance.

|a_e-a_i|t\geq \frac{1}{k} therefore t\geq \frac{1}{k|a_e-a_i|} when k>2. Also, 1-|a_e-a_i|t\geq \frac{1}{k} therefore t\leq \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right). Finally, observing that this is a periodic process with a period of \frac{1}{|a_e-a_i|}, we come upon the conclusion that a_e is lonely as compared to a_i when t\in \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right], where n_i\in\Bbb{N}.

Now we prove the loneliess of a_e with respect to every other runner. This is equivalent to the statement

\bigcap_{i\in A} \left[\frac{1}{k|a_e-a_i|}+\frac{n_i}{|a_e-a_i|}, \left (1-\frac{1}{k}\right )\left(\frac{1}{|a_e-a_i|}\right)\frac{n_i}{|a_e-a_i|}\right], where A=\{1,2,3\dots,k\}\setminus\{e\}. Also, note that all n_i can take different integral values.

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