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LONELY RUNNERS IN FUNCTION FIELDS

Part of: Designs and configurations Probabilistic theory: distribution modulo $1$; metric theory of algorithms Extremal combinatorics Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  12 April 2019

Sam Chow  and
Luka Rimanić
Sam Chow
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, U.K. email [email protected]
Luka Rimanić
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K. email [email protected]
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Abstract

The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.

MSC classification

Primary: 11J71: Distribution modulo one
Secondary: 11K41: Continuous, $p$-adic and abstract analogues 05D05: Extremal set theory 05B20: Matrices (incidence, Hadamard, etc.)
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 677 - 701
Copyright
Copyright © University College London 2019 

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