On the lonely runner conjecture

Pandey, Ram Krishna

doi:10.21136/mb.2010.140683

Cited by 4 publications

(5 citation statements)

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“…Finally, the conjecture for seven runners was established by Barajas and Serra [1] in 2008 and it remains open for all integers n ≥ 8. Several other problems related to the Lonely Runner Conjecture have also been profusely studied such as the gap of loneliness [20,12,13,24] or the validity of the conjecture under various hypotheses on the velocities [19,22,2,17,24]. However, to our knowledge the question of the size of the time required for a runner to get lonely has surprisingly not been really adressed before, this is the purpose of the present paper.…”

Section: Introductionmentioning

confidence: 93%

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On the time for a runner to get lonely

Rifford¹

2021

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The Lonely Runner Conjecture asserts that if n runners with distinct constant speeds run on the unit circle R/Z starting from 0 at time 0, then each runner will at some time t > 0 be lonely in the sense that she/he will be separated by a distance at least 1/n from all the others at time t. In investigating the size of t, we show that an upper bound for t in terms of a certain number of rounds (which, in the case where the lonely runner is static, corresponds to the number of rounds of the slowest non-static runner) is equivalent to a covering problem in dimension n − 2. We formulate a conjecture regarding this covering problem and prove it to be true for n = 3, 4, 5, 6. We also show that the so-called gap of loneliness in one round, where we have m + 1 runners including one static runner, is bounded from below by 1/(2m − 1) for all integer m ≥ 2.

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Section: Introductionmentioning

confidence: 93%

“…Figure 11: If [1, ∞) d K d (δ d )/K 0,N −1 (δ d )does not hold, then there is a point of the form t z z satisfying(19) such that tz / ∈ K d (δ d )/K 0,N −1 (δ d ) for all t ∈ [1, t z )…”

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confidence: 99%

On the time for a runner to get lonely

Rifford¹

2021

Preprint

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“…Many such results impose "lacunarity" conditions on the speeds (i.e., require the speeds to grow at least as fast as a geometric progression). See, e.g., Pandey [18], Ruzsa, Tuza, and Voigt [22] Dubickas [15], and Czerwiński [12]. In a slightly different direction, Tao [23] proved several results on the case where all speeds are small.…”

Section: Introductionmentioning

confidence: 98%

Barely lonely runners and very lonely runners

Kravitz

2019

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We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number x, let x denote the distance from x to the nearest integer. For each set of positive integer speeds v 1 , . . . , vn, we define the associated maximum loneliness to beThe Lonely Runner Conjecture asserts that ML(v 1 , . . . , vn) ≥ 1 n + 1 for all choices of v 1 , . . . , vn. If the Lonely Runner Conjecture is true, then the quantity 1/(n + 1) is the best possible, for there are known equality cases with ML(v 1 , . . . , vn) = 1/(n + 1). A natural but (to our knowledge) hitherto unasked question is:If v 1 , . . . , vn satisfy the Lonely Runner Conjecture but are not an equality case, must ML(v 1 , . . . , vn) be uniformly bounded away from 1/(n + 1)? We conjecture that, contrary to what one might expect, this question has an affirmative answer that reflects an underlying rigidity of the problem. More precisely, we conjecture that for each choice of v 1 , . . . , vn, we have either ML(v 1 , . . . , vn) = s/(ns + 1) for some s ∈ N or ML(v 1 , . . . , vn) ≥ 1/n. Our main results are: confirming this stronger conjecture for n ≤ 3; and confirming it for n = 4 and n = 6 in the case where one speed is much faster than the rest. We also obtain a number of related results.

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“…The work [8] focuses on a specific case for seven runners, using congruences for dividing the track into appropriate intervals. Pandey (2009) [9] proved the conjecture for two or more runners provided the speed of the (𝑖 + 1)-th runner is more than double the speed of the 𝑖-th runner for each 𝑖, with the speeds arranged in an increasing order. Finding a universal conjecture verification for any 𝑛 is still an open problem.…”

Section: Introductionmentioning

confidence: 99%

Solving Lonely Runner Conjecture through differential geometry

Ďuriš

Šumný²,

Gonda

et al. 2022

Journal of Applied Mathematics, Statistics and Informatics

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The Lonely Runner Conjecture is a known open problem that was defined by Wills in 1967 and in 1973 also by Cusick independently of Wills. If we suppose n runners having distinct constant speeds start at a common point and run laps on a circular track with a unit length, then for any given runner, there is a time at which the distance of that runner is at least 1/n from every other runner. There exist several hypothesis verifications for different n mostly based on principles of approximation using number theory. However, the general solution of the conjecture for any n is still an open problem. In our work we will use a unique approach to verify the Lonely Runner Conjecture by the methods of differential geometry, which presents a non-standard solution, but demonstrates to be a suitable method for solving this type of problems. In the paper we will show also the procedure to build an algorithm that shows the possible existence of a solution for any number of runners.

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On the lonely runner conjecture

Cited by 4 publications

References 13 publications

On the time for a runner to get lonely

On the time for a runner to get lonely

Barely lonely runners and very lonely runners

Solving Lonely Runner Conjecture through differential geometry

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