A Variant Of Chaitin's Omega function

Abstract

We investigate the continuous function $f$ defined by $$x\mapsto \sum _{\ddot{I}ƒ{\le }_{L}x}{2}^{-K(\ddot{I}ƒ)}$$ 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 $\hat{I}\text{c◯}$); (iii) the range of $f$ is a null, nowhere dense, perfect ${\hat{I}}_1^0({\varnothing }^{\prime })$ class with Hausdorff dimension $1$; (iv) $f(x)\oplus x{\ge }_T{\varnothing }^{\prime }$ 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.