论文标题
家庭的最大密度和Kappa值$ \ {a，a+1,2a+1，n \} $和$ \ {a，a+1,2a+1,3a+1，n \} $
Maximal density and the kappa values for the families $\{a,a+1,2a+1,n\}$ and $\{a,a+1,2a+1,3a+1,n\}$
论文作者
论文摘要
让$ m $成为一组积极的整数。我们研究了非负整数$ s $的最大密度$μ（m）$，其元素在$ m $中没有差异。 1973年，康托尔（Cantor）和戈登（Gordon）建立了$ | \ m | \ leq 2 $的$μ（m）$的公式。从那时起，许多研究人员一直在处理这个问题，并在此案中发现了一些部分结果，包括$ | m | \ geq 3 $，包括案例中的一些结果，$ m $是无限的集合。在本文中，我们研究了家庭的最大密度问题$ m = \ {a+1,2a+1，n \} $和$ m = \ {a+1,2a+1,3a+1，n \} $，其中$ a $ a $和$ n $是正整数。在大多数情况下，我们找到了由$κ（m）$表示的参数\ textit {kappa}的界限，实际上是$μ（m）$的下限。参数$κ（M）$由于与二聚体近似值中的“孤独跑步者猜想”和诸如图形理论中“圆形着色”和“分数着色”之类的“孤独跑者猜想”的联系，已经具有其重要性。
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 $|M|\leq 2$. 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 $κ(M)$, 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 the problems such as the "lonely runner conjecture" in Diophantine approximations and coloring parameters such as "circular coloring" and "fractional coloring" in graph theory.
