Functions and Analysis

    

Arbitrary Approximation of the Shifted-LRC by the LRC

Authors: Deepak Ponvel Chermakani

Consider an instance of the Shifted Lonely Runner Conjecture (S-LRC) where all n runners (except the stationary runner 0) have integer speeds and start from real values in [0,1[ at time t=0. We show that one can derive an alternative vector of starting points that can be made to be arbitrarily close to the initial vector of starting points. The alternative starting point of each runner i is a rational in [0,1[ and is expressible as (qi / P) where P is a large prime and qi is an integer in [0, P-1]. The S-LRC instance with the alternative starting points, allows a minimal loneliness gap of f, if and only if, the corresponding LRC allows a minimal loneliness gap of f, where f is a desired fraction in ]0,1[. This finding is important in the light of recent counter-examples to the shifted-LRC.

Comments: 2 pages, 2 Theorems.

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Submission history

[v1] 2026-06-20 22:13:49

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