Extending bilipschitz mappings between separated nets

Abstract: We prove that every $L$-bilipschitz mapping $\mathbb{Z}<sup>2\to\mathbb{R}<sup>2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}<sup>2\to\mathbb{R}<sup>2$ and provide an upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}<sup>2$ instead of $\mathbb{Z}<sup>2$. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.