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    What’s newUpdates on my research and expository papers, discussion of open problems, and other maths-related topics.  By Terence TaoHomeAboutCareer adviceOn writingBooksAppletsSubscribe to feedA remark on the lonely runner conjecture13 May, 2015 in expository, math.CA, question | Tags: arithmetic progressions, lonely runner conjecture | by Terence Tao | 17 commentsThe lonely runner conjecture is the following open problem:Conjecture 1  Suppose one has  runners on the unit circle , all starting at the origin and moving at different speeds. Then for each runner, there is at least one time  for which that runner is “lonely” in the sense that it is separated by a distance at least  from all other runners.One can normalise the speed of the lonely runner to be zero, at which point the conjecture can be reformulated (after replacing  by ) as follows:Conjecture 2  Let  be non-zero real numbers for some . Then there exists a real number  such that the numbers  are all a distance at least  from the integers, thus  where  denotes the distance of  to the nearest integer.This conjecture has been proven for , but remains open for larger . The bound  is optimal, as can be seen by looking at the case  and applying the Dirichlet approximation theorem. Note that for each non-zero , the set  has (Banach) density  for any , and from this and the union bound we can easily find  for which for any , but it has proven to be quite challenging to remove the factor of  to increase  to . (As far as I know, even improving  to  for some absolute constant  and sufficiently large  remains open.)The speeds  in the above conjecture are arbitrary non-zero reals, but it has been known for some time that one can reduce without loss of generality to the case when the  are rationals, or equivalently (by scaling) to the case where they are integers; see e.g. Section 4 of this paper of Bohman, Holzman, and Kleitman.In this post I would like to remark on a slight refinement of this reduction, in which the speeds  are integers of bounded size, where the bound depends on . More precisely:Proposition 3  In order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that the  are integers of size at most , where  is an (explicitly computable) absolute constant. (More precisely: if this restricted version of the lonely runner conjecture is true for all , then the original version of the conjecture is also true for all .)In principle, this proposition allows one to verify the lonely runner conjecture for a given  in finite time; however the number of cases to check with this proposition grows faster than exponentially in , and so this is unfortunately not a feasible approach to verifying the lonely runner conjecture for more values of  than currently known.One of the key tools needed to prove this proposition is the following additive combinatorics result. Recall that a generalised arithmetic progression (or ) in the reals  is a set of th
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