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The lonely runner problem for many runners. (English) Zbl 1253.11073

The lonely runner conjecture asserts that, for any positive integer \(n\) and pairwise distinct positive numbers \(v_1<\cdots <v_n\) there is a positive number \(t\) such that the distance \(\| tv_i\|\) of \(tv_i\) to its closest integer is at least \(1/(n+1)\) for each \(i=1,\ldots ,n\). The conjecture originally comes from J. M. Wills [Monatsh. Math. 71, 263–269 (1967; Zbl 0148.27505)] in the context of Diophantine approximation.
Y. Peres and W. Schlag [Bull. Lond. Math. Soc. 42, No. 2, 295–300 (2010; Zbl 1215.05074)] introduced a version of the local Lovász lemma to deal with a closely related problem on lacunary sequences which was then used by the author [Comb. Probab. Comput. 17, No. 3, 339–345 (2008; Zbl 1221.11164)] for a multiple Diophantine approximation. The present paper is an application of the latter result to the lonely runner problem which results in the verification of the conjecture for a sequence satisfying \(v_{i+1}\geq v_i(1+c\log n/n)\) and large enough \(n\) (specifically the author proves the statement for \(c=33\) and \(n\geq 16342\)).

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J25 Diophantine inequalities
11J71 Distribution modulo one