Maximal density and the kappa values for the families $\{a,a+1,2a+1,n\}$ and $\{a,a+1,2a+1,3a+1,n\}$

TL;DR

This paper investigates the maximal density of nonnegative integer sets avoiding differences in specific families of integers, providing bounds for a key parameter linked to longstanding problems in combinatorics and number theory.

Contribution

It introduces bounds for the parameter (M) for particular families of integer sets, extending understanding of the maximal density problem for these cases.

Findings

Bounds for (M) are established for families =(a,a+1,2a+1,n) and =(a,a+1,2a+1,3a+1,n).

Results connect the problem to the lonely runner conjecture and graph coloring parameters.

The study advances the understanding of difference-avoiding sets in additive combinatorics.

Abstract

Let $M$ be a set of positive integers. We study the maximal density $\mu(M)$ 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 $\mu(M)$ for $|M|\leq 2$. Since then, many researchers have worked upon the problem and found several partial results in the case $|M|\geq 3$, including some results in the case, $M$ 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. In most of the cases, we find bounds for the parameter \textit{kappa}, denoted by $\kappa(M)$, which actually serves as a lower bound for $\mu(M)$. The parameter $\kappa(M)$ has already got its importance due to its rich connection with the problems such as the "lonely runner conjecture" in Diophantine approximations and coloring parameters such as "circular coloring" and "fractional coloring" in graph theory.