Randomness and differentiability in higher dimensions

TL;DR

This paper explores the relationship between algorithmic randomness and differentiability in higher dimensions, establishing new theorems that connect computable randomness and weak 2-randomness with differentiability of certain classes of functions.

Contribution

It introduces effective versions of Rademacher's theorem and characterizes weak 2-randomness through differentiability of computable functions of multiple variables.

Findings

Computable randomness implies differentiability of computable Lipschitz functions in multiple dimensions.

Weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions.

Theorems extend classical results to the realm of algorithmic randomness and higher-dimensional analysis.

Abstract

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies differentiability of computable Lipschitz functions of several variables. Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.