Sequences of Integers with Three Missing Separations

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.