Oscillating sequences with subadditivity on shrinking corridors
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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.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis
- Keywords
- subadditivity, Fekete lemma, shrinking corridors, oscillating sequences, submultiplicative growth
- Manuscript date
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
The five-member package contains the strengthened 9-page manuscript, complete self-contained LaTeX source and bibliography, reader instructions, citation metadata and checksums. Build main.tex with Tectonic or two pdflatex runs using the standard packages listed in README.md. The analytic proofs require no computational supplement or external data.
References and related resources
- 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/.