Oscillating sequences with subadditivity on shrinking corridors
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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 ∫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.
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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
Square-integral dichotomy for the monotone counting class and arbitrary-real-sequence convergence with eventual adjacent-pair legality. A positive nonincreasing width with divergent integrals of every finite positive power admits an oscillating integer-valued linearly bounded subadditive sequence when that legality condition fails.
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 eight-member package contains the ten-page manuscript, editable LaTeX, reader instructions, citation metadata, exact finite controls for the missing-adjacent-pair construction and a complete checksum manifest. The analytic proofs establish the general statements; finite controls corroborate the explicit arithmetic construction. See README.md for compilation and reproduction.
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/.