Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}

Abstract

ABSTRACT Let M be a set of positive integers. We study the maximal density μ(M) of the sets of nonnegative integers S whose elements do not differ by an element in M. In 1973, Cantor and Gordon established a formula for μ(M) for |M| ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case |M| ≥ 3, including some results in the case when M is an infinite set. In this paper, we study the maximal density problem for the families M = {a, a+1, 2a+1, n} and M = {a, a+1, 2a+1, 3a+1, n}, where a and n are positive integers and n is sufficiently large. In most of the cases, we find bounds for the parameter kappa, denoted by κ(M), which actually serves as a lower bound for μ(M). The parameter κ(M) has already got its importance due to its rich connection with problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = {a, a +1, 2a + 1,…, (s – 2)a + 1, n}, where s ≥ 5 and mention related problems in the remaining cases for future work.