Polynomial Space Randomness in Analysis

TL;DR

This paper explores the relationship between polynomial space randomness and the Lebesgue differentiation theorem, introducing a new variant called weakly pspace-random points and proving the theorem's validity for these points.

Contribution

It generalizes Ko's framework to define weakly pspace-random points and proves the Lebesgue differentiation theorem holds for them.

Findings

Lebesgue differentiation theorem holds for weakly pspace-random points

Introduces weakly pspace-random points as a new variant of polynomial space randomness

Extends polynomial space computability framework to analyze randomness in analysis

Abstract

We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue differentiation theorem. We generalize Ko's framework for polynomial space computability in $\mathbb{R}^n$ to define \textit{weakly pspace-random} points, a new variant of polynomial space randomness. We show that the Lebesgue differentiation theorem holds for every weakly pspace-random point.