Planar bilipschitz extension from separated nets

We prove that every 𝐿-bilipschitz mapping ℤ 2 → ℝ2 canbe extended to a 𝐶(𝐿)-bilipschitz mapping ℝ2 → ℝ2,and we provide a polynomial upper bound for 𝐶(𝐿).Moreover, we extend the result to every separated netin ℝ2 instead of ℤ 2, with the upper bound gaininga polynomial dependence on the separation and netconstants associated to the given separated net. Thisanswers an Oberwolfach question of Navas from 2015and is also a positive solution of the two-dimensionalform of a decades old open (in all dimensions at leasttwo) problem due to Alestalo Trotsenko and Väisälä.