Luzin's (N) and randomness reflection

TL;DR

This paper characterizes when computable functions reflect various forms of algorithmic randomness, linking classical analysis properties like Luzin's (N) to the reflection of different randomness notions.

Contribution

It establishes equivalences between Luzin's (N) property and reflection of multiple randomness notions, connecting real analysis with algorithmic randomness.

Findings

Luzin's (N) property iff reflection of $oldsymbol{ m ext{Pi}^1_1}$-randomness

Luzin's (N) property iff reflection of $oldsymbol{ m ext{Delta}^1_1( ext{O})}$-randomness

Luzin's (N) property iff reflection of $ ext{O}$-Kurtz randomness

Abstract

We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $\Pi^1_1$-randomnes, 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-L\"of 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.