A Variant Of Chaitin's Omega function
Year: 2025
Type: preprint
Abstract: We investigate the continuous function $f$ defined by $$x\mapsto \sum_{σ\le_L x }2^{-K(σ)}$$ 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 $Ω$); (iii) the range of $f$ is a null, nowhere dense, perfect $Π^0_1(\emptyset')$ 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.
Source: arXiv (Cornell University)
Authors: Li, Yuxuan, Zhang, Shuheng, Zhang, Xiaoyan, Zhao, Xuanheng
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077905291
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