Abstract:
This thesis studies connections between computable randomness in Rn and various properties related to differentiability. This research was inspired by results in two recent papers: one by Brattka, Miller and Nies and another one by Freer, Kjos-Hanssen, Nies and Stephan. In those papers it was shown that computable randomness on the unit interval can be characterized by differentiability properties of computable Lipschitz functions and computable monotone function. We generalize to Rn most of those results. Moreover, we show several new results of this kind both on the real line and on Rn. In the process, we prove effective versions of several notable classical results such as: Rademacher’s theorem, Aleksandrov’s theorem, Sard’s theorem for monotone Lipschitz functions and Brenier’s theorem. In most cases we prove converse results as well.