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The Lonely Runner Conjecture, Linear Relations of Speeds

Fosheng Wang

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Let  be an integer and let  be distinct nonzero real numbers (also called speeds). The Lonely Runner Conjecture states that there exists  such that  for all . Viewing  as an abelian group, we prove that the conjecture holds for speeds  if the speeds are linearly independent or “almost linearly independent” in the group , and we give a measure of “how lonely the runners are” in these cases.


Mathematics Subject Classification: 11B75.



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