Research Article · Preprint Mathematics
A square-integral threshold for subadditivity on shrinking corridors
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 3 · Permanent link
Overview
Abstract
Let w:[T,∞)→[0,1] be measurable and nonincreasing. For exact eventual subadditivity on the integer corridor n≤m≤n+n w(log n), we identify a square-integral threshold. If ∫_T^∞ w(s)² ds diverges, normalized slopes converge for nondecreasing nonnegative linearly bounded sequences, without continuity or an adjacent-pair assumption. In this divergent-integral case, if adjacent integer pairs are eventually admissible, a prefix-envelope and dyadic-record argument removes all monotonicity, sign and growth hypotheses: every real sequence satisfying the corridor inequality has a normalized limit in ℝ∪{−∞}. In particular, n/√log n is a sufficient sublinear corridor for arbitrary real sequences. If the integral is finite, a shifted-tail correction gives strictly increasing integer submultiplicative and supermultiplicative counterexamples with logarithmic growth endpoints 1.99 and 2.01. Thus the criterion is an exact dichotomy in the monotone counting class, and also for arbitrary real sequences when the adjacent-pair condition holds. Without monotonicity, adjacent-pair legality cannot simply be omitted: we construct a positive nonincreasing width for which every finite positive-power integral diverges, yet an integer-valued linearly bounded subadditive sequence has normalized lower and upper limits zero and one.
Keywords subadditivity · Fekete lemma · shrinking corridors · oscillating sequences · submultiplicative growth
Scope of the work
The square-integral corridor threshold governs the stated integer subadditive convergence problem, including monotone and unrestricted assertions, rounding controls and logarithmic-width examples.
Reproduction notes
The 11-page manuscript and editable LaTeX sources are supplied in this 8-member package. README.md gives build instructions. verify.py provides the finite mathematical controls specified in the manuscript alongside its analytic proofs.
The manuscript
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Author disclosures
AI systems assisted research exploration, literature discovery, drafting and internal checking. The author retains responsibility for the manuscript. Internal AI-assisted and computational checks are distinct from external human peer review.
References and related publications
- Z. Füredi and I. Z. Ruzsa, Nearly subadditive sequences , Acta Math. Hungar. 161 (2020), 401– 411. doi:10.1007/s10474-020-01071-0. Question attribution and mathematical comparisons use arXiv:1810.11723v1, Section 7.
- A. Kulikov and F. Shao, Fekete’s lemma in Banach spaces , arXiv:2411.17380v1 (2024).
- F. Shao and contributors, High dimensional Fekete’s subadditive lemma , MathOverflow discussion (2023), https://mathoverflow.net/questions/459773/.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). A square-integral threshold for subadditivity on shrinking corridors (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/oscillating-sequences-subadditivity-shrinking-corridors/v/3
BibTeX
@techreport{luminary_oscillating_sequences_subadditivity_shrinking_corridors_v3,
title = {A square-integral threshold for subadditivity on shrinking corridors},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 3},
url = {https://research.luminaryoddities.com/oscillating-sequences-subadditivity-shrinking-corridors/v/3}
}Loading discussion…