AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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For fixed $d\ge 1$, let $t_{n,d}$ be the number of subsets of$[n]^d$ that tile $\mathbb{Z}^d$ by translations. We prove that\[t_{n,d}=(3^{1/3})^{n^d\pm o(n^d)}.\]
A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining t...
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