A square-integral threshold for subadditivity on shrinking corridors
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Abstract
Let w be a measurable nonincreasing relative width. For exact eventual integer subadditivity on n≤ m≤ n+n w(log n), we prove a sharp square-integral dichotomy in the nonnegative nondecreasing linearly bounded class. Divergence of ∫ w^2 forces normalized-slope convergence, including widths with jumps and missing adjacent pairs. With eventual adjacent-pair legality, convergence holds for arbitrary real sequences, with limit in ℝ ∪ {−∞}. Thus n/sqrt(log n) is a sufficient sublinear corridor. We further replace adjacent legality by balanced legal predecessors of gap O(N/[(log N)^(3/2)(log log N)^η]), η>1/2. Growing local maxima preserve a smaller literal corridor, and dyadic propagation transfers their limit to the original sequence. A staircase example satisfies this condition while failing every bound O(N/(log N)^(3/2+ε)), ε>0, and every fixed sublinear power bound. When ∫ w^2 is finite, strictly increasing integer submultiplicative and supermultiplicative sequences can have two distinct exponential rates. Conversely, isolated outputs permit oscillation even when every finite positive-power integral of w diverges. These results distinguish integrated width from integer connectivity.
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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
- none
Scope
Square-integral width dichotomy for literal shrinking integer corridors; arbitrary-real convergence under adjacent legality or balanced legal-predecessor gaps O(N/((log N)^(3/2)(log log N)^η)), η > 1/2. Integer connectivity is distinguished from integrated width; mere representability is not claimed sufficient.
Disclosures
AI-assisted tools supported exploration, internal checking and manuscript preparation.
Reproduction
The 18-page manuscript and complete editable source are supplied in a 9-member public package. README.md gives build instructions, dependencies and reproduction scope. Build reproduction is qualified by the compiler, support files and timestamp settings specified in the supplied instructions. The mathematical conclusions have analytic proofs; accompanying finite checks, where present, retain the limited scopes stated in the package.
References and related resources
- Z. Füredi and I. Z. Ruzsa. Nearly subadditive sequences. Acta Mathematica Hungarica 161 (2020), 401-411. Comparison and question: arXiv:1810.11723v1, Section 7.
- S. Schubr. Fekete convergence from balanced integer predecessors in shrinking corridors. Luminary Research Archive, edition 2 (2026).
- 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).
- N. G. de Bruijn and P. Erdős. Some linear and some quadratic recursion formulas. II. Proc. Koninkl. Nederl. Akad. Wetensch. A 55 (1952), 152-163.
- C. J. K. Batty, M. J. Pelling and D. G. Rogers. Some recurrence relations of recursive minimization. SIAM Journal on Algebraic Discrete Methods 3 (1982), 13-29.
- C. J. K. Batty and D. G. Rogers. Some maximal solutions of the generalized subadditive inequality. SIAM Journal on Algebraic Discrete Methods 3 (1982), 369-378.
- R. J. Wallace. Optimal solutions of restricted subadditive inequalities. Bulletin of the Australian Mathematical Society 38 (1988), 475-477. Author’s thesis summary.