Research Article · Preprint Mathematics
Oscillating sequences with subadditivity on shrinking corridors
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
For measurable nonincreasing relative widths w, this paper identifies a square-integral threshold for exact eventual subadditivity on the integer corridor n ≤ m ≤ n + nw(log n). Divergence of ∫w² forces normalized-slope convergence for nondecreasing nonnegative linearly bounded sequences, without continuity or an adjacent-pair assumption. If adjacent integer pairs are eventually admissible, a prefix-envelope and dyadic-record argument removes the monotonicity, sign and growth hypotheses: every real sequence obeying the corridor inequality has a normalized limit in ℝ ∪ {−∞}. In particular, n/√log n is a sufficient sublinear corridor for arbitrary real sequences. For a finite square integral, a shifted-tail correction produces strictly increasing integer submultiplicative and supermultiplicative counterexamples with logarithmic growth endpoints 1.99 and 2.01. The resulting dichotomy is exact in the monotone counting class and, with the adjacent-pair condition, for arbitrary real sequences. Divergent widths with missing adjacent pairs remain unresolved in the unrestricted real-sequence setting.
Keywords subadditivity · Fekete lemma · shrinking corridors · oscillating sequences · submultiplicative growth
Scope of the work
The square-integral dichotomy is exact for the stated monotone counting class. The unrestricted real-sequence convergence theorem also requires eventual admissibility of adjacent integer pairs. No conclusion is asserted for general real sequences in the divergent-integral case with missing adjacent pairs.
Reproduction notes
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Author disclosures
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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). Oscillating sequences with subadditivity on shrinking corridors (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/oscillating-sequences-subadditivity-shrinking-corridors/v/1
BibTeX
@techreport{luminary_oscillating_sequences_subadditivity_shrinking_corridors_v1,
title = {Oscillating sequences with subadditivity on shrinking corridors},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/oscillating-sequences-subadditivity-shrinking-corridors/v/1}
}Loading discussion…