[Paper Review] Variants on the Berz sublinearity theorem

Ν. H. Bingham, A. J. Ostaszewski|arXiv (Cornell University)|Dec 14, 2017

Advanced Topology and Set Theory44 references3 citations

TL;DR

This paper presents variants of the Berz sublinearity theorem using only the Axiom of Dependent Choices (DC) instead of the full Axiom of Choice (AC), establishing conditions under which sublinear functions on the reals are linear. It introduces 'thinned' versions where homogeneity and subadditivity conditions are restricted to dense additive subgroups, with continuity and boundedness ensuring linearity.

ABSTRACT

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

Motivation & Objective

• To establish a group-theoretic analogue of the Hahn-Banach Extension Theorem using only DC, avoiding reliance on full AC.
• To investigate whether the universal quantifier in Berz’s sublinearity condition (i.e., N-homogeneity for all real x) can be weakened to a dense additive subgroup A ⊆ ℝ.
• To provide conditions under which a subadditive, N-homogeneous function on a dense subgroup A extends to a linear function on ℝ, using only DC.
• To explore the role of continuity and local boundedness in ensuring linearity under quantifier-weakened conditions.
• To connect these results to classical subadditivity, regular variation, and the Kingman subadditive ergodic theorem via axiomatic and topological methods.

Proposed method

• Utilizes Theorem 0 (Goldie’s continuity criterion) to show that subadditive functions continuous at 0 are continuous everywhere, provided they vanish at 0.
• Applies Theorem 0+ (Kingman’s Baire Category Theorem-based bounding) to establish linear majorants for subadditive functions under mild regularity conditions.
• Imposes local boundedness and subadditivity on a function S defined on ℝ, then shows that continuity at 0 implies global continuity and linearity on ℝ₊ and ℝ₋.
• Introduces 'thinning' by restricting the domain of N-homogeneity to a dense additive subgroup A ⊆ ℝ, while maintaining subadditivity and local boundedness.
• Uses asymptotic analysis of S(t)/t near 0 and ∞ to derive necessary and sufficient conditions for linearity when homogeneity is restricted to A.
• Relies on DC instead of AC to prove extension theorems, avoiding strong choice principles while preserving key results in sublinearity and homomorphism extension.

Experimental results

Research questions

• RQ1 Can the Berz sublinearity theorem be proven using only the Axiom of Dependent Choices (DC) rather than the full Axiom of Choice (AC)?
• RQ2 Under what conditions does a subadditive, N-homogeneous function on a dense additive subgroup A ⊆ ℝ extend to a linear function on ℝ?
• RQ3 What side-conditions are necessary and sufficient to ensure that weakening the universal quantifier in N-homogeneity (from all x ∈ ℝ to x ∈ A) still yields linearity?
• RQ4 How do the results relate to classical subadditivity theorems, such as those of Hille-Phillips and Kolmogorov, under DC?
• RQ5 Can the Hahn-Banach-type extension theorems for additive functions be established with only DC, particularly in separable or amenable group settings?

Key findings

• A sublinear function S: ℝ → ℝ that is locally bounded is continuous everywhere and linear on ℝ₊ and ℝ₋, using only DC.
• Theorem 3 establishes a DC-based extension of the Hahn-Banach theorem for additive functions, avoiding reliance on AC or PIT.
• For a subadditive function S: ℝ → ℝ with S(0) = 0, continuity at 0 is equivalent to S(zₙ) → 0 for some sequence zₙ ↗ 0.
• When N-homogeneity is restricted to a dense additive subgroup A ⊆ ℝ and S is locally bounded, S is linear on ℝ if and only if the asymptotic ratio S(t)/t has matching limits at 0 and ∞.
• The results generalize to functions with values in normed spaces, where S(x) = ||x||S(uₓ) for unit vectors uₓ, with uniform boundedness of S(uₓ) over x ≠ 0.
• The paper shows that the separable case of Badura’s group-theoretic Hahn-Banach result may be provable with only DC, leaving the role of completeness open.