Rectifying separated nets. (English) Zbl 1165.26007

Burago, D.; Kleiner, B.

Geom. Funct. Anal. 12, No. 1, 80-92 (2002).

From the text: Suppose that \(Y\subset\mathbb R^2\) is a separated net. For \(\rho>0\) and each measurable subset \(U\subset\mathbb R^2\), define \(e_\rho(U)\) to be the density deviation
\[ \max\bigg( \frac{\rho|U|}{\#(U\cap Y)}, \frac{\#(U\cap Y)}{\rho|U|} \bigg). \]
Then define \(E_\rho:\mathbb N\to\mathbb R\) by letting \(E_\rho(k)\) be the supremum of the quantities \(e_\rho(U)\), where \(U\) ranges over all squares of the form \([i,i+k]\times [j,j+k]\) for \(i,j\in\mathbb Z\). If there exists a \(\rho>0\) such that the product \(\prod_m E_\rho(2^m)\) converges, then \(Y\) is bi-Lipschitz to \(\mathbb Z^2\).

MSC:

26B35	Special properties of functions of several variables, Hölder conditions, etc.
26B10	Implicit function theorems, Jacobians, transformations with several variables
58C07	Continuity properties of mappings on manifolds

Keywords:

bi-Lipschitzean; homeomorphism; Jacobian; density estimates