A square-integral threshold for subadditivity on shrinking corridors
Loading original first page…
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.
Read the full paper in the original PDF.
Edition files
- main.pdf104.4 KiB · pdf
SHA-256
4646f3706a87347e4590f00f52fec679470442d424162cb1b02d483c21ea8d36 - Shrinking_corridor_publication_package.zip117.2 KiB · package
SHA-256
2ac5cc51ec3af35df470e49f37e6af583e6335ea345f3027ffeb4e96431ce572
Available editions
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 corridor threshold governs the stated integer subadditive convergence problem, including monotone and unrestricted assertions, rounding controls and logarithmic-width examples.
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 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.
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/.