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LUZIN’S (N) AND RANDOMNESS REFLECTION

Part of: Real functions Computability and recursion theory

Published online by Cambridge University Press:  30 October 2020

ARNO PAULY ,
LINDA WESTRICK  and
LIANG YU
ARNO PAULY
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF SWANSEASWANSEA, WALES, UKE-mail:arno.m.pauly@gmail.com
LINDA WESTRICK
Affiliation:
DEPARTMENT OF MATHEMATICS PENN STATE UNIVERSITY UNIVERSITY PARK, PA, USE-mail:westrick@psu.edu
LIANG YU
Affiliation:
DEPARTMENT OF MATHEMATICS NANJING UNIVERSITYNANJING CITY, CHINAE-mail:yuliang.nju@gmail.com
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Abstract

We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property (N) if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1({\mathcal {O}})$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever $f(x)$ is R-random, then x is R-random as well. If additionally f is known to have bounded variation, then we show f has Luzin’s (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset '$ -Kurtz randomness. This links classical real analysis with algorithmic randomness.

Keywords

higher randomnessLuzin’s (N)computable analysisrandomness reflection

MSC classification

Primary: 03D32: Algorithmic randomness and dimension 26A30: Singular functions, Cantor functions, functions with other special properties
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 802 - 828
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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