Sequences of integers with three missing separations

Let D be a set of positive integers. We study the maximum density μ(D) of integral sequences in which the separation between any two terms does not fall in D. The sets D considered in this article are mainly of the form {1,j,k}. The closely related function κ(D), the parameter involved in the “lonely runner conjecture,” is also investigated. Exact values of κ(D) and μ(D) are found for many families of D={1,j,k}. In particular, we prove that the...