AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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Let $h\geq2$ be an integer and let $1/h<\theta\leq1/(h-1)$. In this paper, we prove that there exists a strongly minimal asymptotic basis $A$ of order $h$ such that $A(x)\asymp x^\theta$. This solves a problem posed by Nathanson in 1988.
Let $h\geq 3$ be an integer and $0<\alpha<1/h$. In this paper, we prove that there exists a minimal asymptotic basis of order $h$ with asymptotic density $\alpha$. This solves an open problem posed by Erd\H{o}s and Nathanson in 1988.
Let $\mathbb{N}_0=\{0,1,2,\ldots\}$ and let $h\ge2$ be an integer. For a set $A\subseteq\mathbb{N}_0$, write $hA$ for the set of all sums of $h$, not necessarily distinct, elements of $A$. In this paper, we prove that for every $h\ge2$, there is a partition $\mathbb{N}_0=A\sqcup B$ such that $A$ is a minimal asymptotic...
Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certai...
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