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Separated nets in Euclidean space and Jacobians of biLipschitz maps. (English) Zbl 0902.26004

A subset \(X\) of a metric space \(Z\) is a separated net if there are constants \(a,b> 0\) such that \(d(x,x')> a\) for every pair \(x,x'\in X\) and \(d(z,X)< b\) for every \(z\in Z\).
The authors show the existence of separated nets in the Euclidean plane which are not bi-Lipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a bi-Lipschitz map.

MSC:

26B10 Implicit function theorems, Jacobians, transformations with several variables
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