Sequences of Integers with Three Missing Separations

TL;DR

This paper investigates the maximum density of integer sequences avoiding certain separations, providing exact values for specific cases and addressing conjectures related to the lonely runner problem.

Contribution

It introduces new exact values for the functions nd or sets of the form nd nd proves the sharpness of boundary conditions in previous results.

Findings

Exact values of nd or specific nd ases.

Sharpness of boundary conditions in earlier theorems.

Counterexamples to recent conjectures.

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.