Genericity and UD-random reals

Abstract

Avigad introduced the notion of UD-randomness based in Weyl's 1916 definition of uniform distribution modulo one. We prove that there exists a weakly 1-random real that is neither UD-random nor weakly 1-generic. We also show that no 2-generic real can Turing compute a UD-random real.

Key developments in algorithmic randomness

“…Since the publication of the two monographs due to Nies [81] and Downey and Hirschfeldt [26], a majority of research in algorithmic randomness (at least outside of the setting of resource-bounded randomness) has involved notions of randomness that fall between Kurtz randomness and 2-randomness when measured in terms of strength. One exception is UD-randomness, introduced by Avigad [3] and further developed by Calvert and Franklin [14]. Based on the concept of uniform distribution studied by Weyl [106], UD-randomness is implied by Schnorr randomness but incomparable with Kurtz randomness, as shown by Avigad. In particular, there has been a great deal of interest in notions even stronger than Martin-Löf randomness.…”

Algorithmic randomness and Fourier analysis

“…To date, results of this nature have been discovered in ergodic theory [4,19,20,22,26,36,50], differentiability [5,6,21,27,36,40,43], Brownian motion [1,2,18], and other topics in analysis [3,9,45]. In this paper, we add Fourier series to this list by considering Carleson's Theorem.…”