Computable randomness and monotonicity

TL;DR

This paper establishes a characterization of computable randomness in n-dimensional real space through the differentiability of all computable monotone functions at a point.

Contribution

It provides a novel equivalence between computable randomness and the differentiability of computable monotone functions in multiple dimensions.

Findings

Computably random points are exactly those where all computable monotone functions are differentiable.

The result extends classical differentiability characterizations to the computability setting.

This bridges computability theory and geometric analysis in higher dimensions.

Abstract

We show that $z\in\R^n$ is computably random if and only if every computable monotone function on $\R^n$ is differentiable at $z$.