This book is concerned with the theory of computability and complexity over the real
numbers. This theory was initiated by Turing, Grzegorczyk, Lacombe, Banach and Mazur …

Randomness and differentiability Page 1 TRANSACTIONS OF THE AMERICAN
MATHEMATICAL SOCIETY Volume 368, Number 1, January 2016, Pages 581–605 http://dx.doi.org/10.1090/tran/6484 …

We report on some recent work centered on attempts to understand when one set is more
random than another. We look at various methods of calibration by initial segment …

One approach to understanding the fine structure of initial segment complexity was
introduced by Downey, Hirschfeldt and LaForte. They define $ X\leq _K Y $ to mean that …

It is time for a new paper about open questions in the currently very active area of
randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 …

Let R be a notion of algorithmic randomness for individual subsets of ℕ. A set B is a base for
R randomness if there is a Z≥ TB such that Z is R random relative to B. We show that the …

In this paper, we define new notions of randomness based on the difference hierarchy. We
consider various ways in which a real can avoid all effectively given tests consisting of $ n …

We study the randomness properties of reals with respect to arbitrary probability measures
on Cantor space. We show that every non-computable real is non-trivially random with …

The Jordan decomposition theorem states that every function of bounded variation can be
written as the difference of two non-decreasing functions. Combining this fact with a result of …

We call A weakly low for K if there is ac such that KA (σ)≥ K (σ)− c for infinitely many σ; in
other words, there are infinitely many strings that A does not help compress. We prove that A …