Polynomial Space Randomness in Analysis

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.

💡 Research Summary

The paper investigates the relationship between polynomial‑space (pspace) randomness and a cornerstone result of real analysis, the Lebesgue Differentiation Theorem. The authors first extend Ko’s framework for polynomial‑space computability from the real line to ℝⁿ, defining polynomial‑space L¹‑computable functions as those that can be uniformly approximated by a sequence of simple step functions whose parameters (dyadic endpoints and rational values) are computable within polynomial space. This class coincides with Ko’s notion of pspace‑approximable functions and serves as the analytic objects of interest.
Traditional resource‑bounded randomness is usually defined via martingales, but the authors argue that such definitions are ill‑suited for analysis because they do not interact well with open‑cover arguments. Consequently, they introduce a new variant called weakly pspace‑random points. A weak pspace‑random point is one that passes every pspace W‑test: an effectively presented sequence of open sets {Uₘ} with μ(Uₘ) ≤ 2⁻ᵐ, together with a uniformly pspace‑computable array {S_{k,m}} that approximates each Uₘ from inside (Uₘ ⊆ lim infₖ S_{k,m}). The approximation requirement replaces the exact enumeration used in classical martingale tests, making the notion compatible with measure‑theoretic arguments. Lemma 4 shows that ordinary pspace randomness (no pspace‑computable martingale succeeds) implies weak pspace randomness, so the new notion is strictly weaker but still robust.
The central theorem states that a point x ∈