A Variant Of Chaitin's Omega function

TL;DR

This paper studies a function related to Chaitin's Omega, exploring its differentiability, randomness properties, and computational complexity, revealing deep connections with algorithmic randomness and computability theory.

Contribution

It introduces and analyzes a new variant of Chaitin's Omega, establishing its differentiability, randomness characteristics, and Turing degree properties.

Findings

f is differentiable exactly at density random points

f(x) is x-random iff x is low for Omega

Range of f has Hausdorff dimension 1

Abstract

We investigate the continuous function $f$ defined by $$x\mapsto \sum_{\sigma\le_L x }2^{-K(\sigma)}$$ as a variant of Chaitin's Omega from the perspective of analysis, computability, and algorithmic randomness. Among other results, we obtain that: (i) $f$ is differentiable precisely at density random points; (ii) $f(x)$ is $x$-random if and only if $x$ is weakly low for $K$ (low for $\Omega$); (iii) the range of $f$ is a null, nowhere dense, perfect $\Pi^0_1(\emptyset')$ class with Hausdorff dimension $1$; (iv) $f(x)\oplus x\ge_T\emptyset'$ for all $x$; (v) there are $2^{\aleph_0}$ many $x$ such that $f(x)$ is not 1-random; (vi) $f$ is not Turing invariant but is Turing invariant on the ideal of $K$-trivial reals. We also discuss the connection between $f$ and other variants of Omega.