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André Nies

Publications and source records attributed to André Nies.

23 records · Page 2Linked to original sources
arXiv2014

Logic Blog 2013

The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of $Π^1_1$ randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For instance, what is the strength of Jordan's theorem in analysis? (His theorem states that each function of bounded variation is the difference of two nondecreasing functions.) 3. Randomness and computable analysis. This focusses on the connection of randomness of a real $z$ and Lebesgue density of effectively closed sets at $z$. 4. Exploring similarity relations for Polish metric spaces, such as isometry, or having Gromov-Hausdorff distance $0$. In particular their complexity was studied. 5. Various results connecting computability theory and randomness.

math.LO↗
arXiv2014

Randomness and Differentiability

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function [0,1]->R is differentiable at z. (2) We prove that a real number z in [0,1] is weakly 2-random if and only if each almost everywhere differentiable computable function [0,1]->R is differentiable at z. (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z is ML random if and only if every computable function of bounded variation is differentiable at z, and similarly for absolutely continuous functions. We also use our analytic methods to show that computable randomness of a real is base invariant, and to derive other preservation results for randomness notions.

math.LO↗
arXiv2014

Logic Blog 2011

This year's logic blog has focussed on: 1. Demuth randomness 2. traceability 3. The connection of computable analysis and randomness 4. $K$-triviality in metric spaces.

math.LO↗
arXiv2014

Algorithmic aspects of Lipschitz functions

We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Beyond these principal results, we show that a real z is Schnorr random if and only if every Lipschitz function with $L_1$-computable derivative is differentiable at z.

math.LO↗