18 research outputs found
Random runners are very lonely
Suppose that runners having different constant speeds run laps on a
circular track of unit length. The Lonely Runner Conjecture states that, sooner
or later, any given runner will be at distance at least from all the
other runners. We prove that, with probability tending to one, a much stronger
statement holds for random sets in which the bound is replaced by
\thinspace . The proof uses Fourier analytic methods. We also
point out some consequences of our result for colouring of random integer
distance graphs
Correlation among runners and some results on the Lonely Runner Conjecture
The Lonely Runner Conjecture was posed independently by Wills and Cusick and
has many applications in different mathematical fields, such as diophantine
approximation. This well-known conjecture states that for any set of runners
running along the unit circle with constant different speeds and starting at
the same point, there is a moment where all of them are far enough from the
origin. We study the correlation among the time that runners spend close to the
origin. By means of these correlations, we improve a result of Chen on the gap
of loneliness and we extend an invisible runner result of Czerwinski and
Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a
weak version of the conjecture thus providing some new results.Comment: 18 page
The lonely runner problem for many runners
The lonely runner conjecture asserts that for any positive integer n and any positive numbers v1 < ... < vn there exists a positive number t such that ||vi t|| ≥ 1/(n+1) for every i=1, ...,n. We verify this conjecture for n ≥ 16342 under assumption that the speeds of the runners satisfy vj+1/vj ≥ 1+33 log n/n for j=1, ...,n-1
Lonely runners in function fields
The lonely runner conjecture, now over fifty years old, concerns the
following problem. On a unit length circular track, consider 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 distance at least from the others. We formulate a function field
analogue, and give a positive answer in some cases in the new setting
The lonely runner with seven runners
Suppose runners having nonzero constant speeds run laps on a
unit-length circular track starting at the same time and place. A runner is
said to be lonely if she is at distance at least along the track to
every other runner. The lonely runner conjecture states that every runner gets
lonely. The conjecture has been proved up to six runners (). A
formulation of the problem is related to the regular chromatic number of
distance graphs. We use a new tool developed in this context to solve the first
open case of the conjecture with seven runners
A few more Lonely Runners
Lonely Runner Conjecture, proposed by J\"{o}rg M. Wills and so nomenclatured
by Luis Goddyn, has been an object of interest since it was first conceived in
1967 : Given positive integers and there exists a
positive real number such that the distance of to the nearest
integer is at least , . In a recent
article Beck, Hosten and Schymura described the Lonely Runner polyhedron and
provided a polyhedral approach to identifying families of lonely runner
instances. We revisit the Lonely Runner polyhedron and highlight some new
families of instances satisfying the conjecture. In addition, we relax the
sufficiency of existence of an integer point in the Lonely Runner polyhedron to
prove the conjecture. Specifically, we propose that it suffices to show the
existence of a lattice point of certain superlattices of the integer lattice in
the Lonely Runner polyhedron