Lipschitz mappings of discrete sets

In this thesis we consider Feige's question of whether there always exists a constantly Lipschitz bijection of an n2 -element set S ⊂ Z2 onto a regular lattice of n × n points in Z2 . We propose a solution of this problem in case the points of the set S form a long rectangle or they are arranged in the shape of a square without a part of its interior points. The main part is a summary of Burago's and Kleiner's article [2] and the article by McMullen [12] dealing with the problem of existence of separated nets in R2 that are not bi-Lipschitz equivalent to the integer lattice. This problem looks similar to Feige's problem. According to these articles we construct a separated net that is not bi-Lipschitz equivalent to the integer lattice, using a positive bounded measurable function that is not the Jacobian of a bi-Lipschitz homeomorphism almost everywhere. We present McMullen's construction of such a function and we complete his proof of its correctness.