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arXiv · 1611.03940

Sequences of Integers with Three Missing Separations

Daphne Der-Fen Liu·Grant Robinson
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Abstract

Fix a set $D$ of positive integers. We study the maximum density $\mu(D)$ of sequences of integers in which the separation between any two terms does not fall in $D$. The $D$-sets considered in this article are of the form $\{1,j,k\}$. The closely related function $\kappa(D)$, the parameter involved in the "lonely runner conjecture," is also investigated. Exact values of $\kappa(D)$ and $\mu(D)$ are found for some families of $D=\{1,j,k\}$. We prove that the boundary conditions in two earlier results of Haralambis are sharp. Consequently, our results declaim two conjectures posted recently, and extend some results by Gupta.

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Daphne Der-Fen Liu, Grant Robinson. 2016-11-12. Sequences of Integers with Three Missing Separations. https://arxiv.org/abs/1611.03940

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