Abstract
The Lonely Runner Conjecture, posed independently by Wills and by Cusick, 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 time 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. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing a new result related to the invisible runner theorem of Czerwinski and Grytczuk.
Highlights
The Lonely Runner Conjecture was posed independently by Wills [22] in 1967 and Cusick [10] in 1982
The Lonely Runner Conjecture, posed independently by Wills and by Cusick, 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 time where all of them are far enough from the origin
We introduce dynamic interval graphs to deal with a weak version of the conjecture providing a new result related to the invisible runner theorem of Czerwinski and Grytczuk
Summary
The Lonely Runner Conjecture was posed independently by Wills [22] in 1967 and Cusick [10] in 1982. This result was improved by Chen [7] who showed that, for every set of n nonzero speeds, there exists a time t ∈ R such that tvi. If 2n − 3 is a prime number, the previous result was extended by Chen and Cusick [8] In this case, these authors proved that, for every set of n speeds, there exists a time t ∈ R such that tvi. The last inequality holds under a natural density condition on the set of speeds which covers the more difficult cases where the speeds grow slowly Another interesting result on the Lonely Runner Conjecture, was given by Czerwinski and Grytczuk [13].
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