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arXiv 1912.09355math.CO
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数值半群上数值集的稠密性

Density of Numerical sets associated to a Numerical semigroup

Deepesh Singhal, Yuxin Lin

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摘要：

数值集是包含零的自然数的余有限子集。它的Frobenius数是它的补码中最大的数。每个数值集都有一个相关联的半群$A(T)=\{t\mid t+T\subseteq T\}$，它与$T$具有相同的Frobenius数。对于固定的Frobenius数$f$，存在$2^{f-1}$数值集。已知存在接近于$0.484$的数$γ$，使得映射到$N_f=\{0\}\cup(f,\infty)$的这些数值集的比率渐近地为$γ$。我们确定了一个集合的家庭$N(D,f)$的数值半群，使得对于一个固定的$D$的比率的$2^{f-1}$数值集映射到$N(D,f)$收敛到一个正极限，因为$f$走向无穷大。我们将极限表示为$γ_D$，这些常数之和为$1$，这意味着它们渐近地解释了几乎所有的数值集。

英文摘要：

A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the same Frobenius number as $T$. For a fixed Frobenius number $f$ there are $2^{f-1}$ numerical sets. It is known that there is a number $γ$ close to $0.484$ such that the ratio of these numerical sets that are mapped to $N_f=\{0\}\cup(f,\infty)$ is asymptotically $γ$. We identify a collection of families $N(D,f)$ of numerical semigroups such that for a fixed $D$ the ratio of the $2^{f-1}$ numerical sets that are mapped to $N(D,f)$ converges to a positive limit as $f$ goes to infinity. We denote the limit as $γ_D$, these constants sum up to $1$ meaning that they asymptotically account for almost all numerical sets.

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