Knights are 24/13 times faster than the king

Abstract

On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime $b>a\in {\mathbb{Z}}_{\ge 1}$ such that $a+b$ is odd, define the $(a,b)$-knight and the king as \begin{equation*} \begin{aligned} \mathrm{N}_{a,b} = \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\},\newline \mathrm{K}=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} \subseteq \mathbb{Z}^2, \end{aligned} \end{equation*} respectively. One way to formulate this question is by asking for the average ratio, for $\mathbf{p}\in {\mathbb{Z}}^2$ in a box, between $\min \{h\in {\mathbb{Z}}_{\ge 1}|\mathbf{p}\in h\mathrm{N}\}$ and $\min \{h\in {\mathbb{Z}}_{\ge 1}|\mathbf{p}\in h\mathrm{K}\}$, where $hA=\{{\mathbf{a}}_1+\cdots +{\mathbf{a}}_h|{\mathbf{a}}_1,\dots ,{\mathbf{a}}_h\in A\}$ is the $h$-fold sumset of $A$. We show that this ratio equals $2(a+b)b^2/(a^2+3b^2)$.

7 pages, 2 figures. Fixed typos