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Rectifying separated nets. (English) Zbl 1165.26007
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
Full Text:
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