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Research Article · Preprint Mathematics

Exponential growth of sparse Euler products and normal graded domains

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Algebra
  • Combinatorics
Primary subject
Mathematics
Subject subfields
Algebra · Combinatorics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

Let dᵢ be strictly increasing positive integer degrees and let Mᵢ be positive integers with log(Mᵢ)/dᵢ → δ ∈ (0, ∞ ). For the Euler product with Mᵢ independent polynomial generators of degree dᵢ and one degree-one generator, we prove that the upper logarithmic coefficient rate is δ and the lower rate is max{δ − θ, 0}, where θ = lim supᵢ log(dᵢ₊₁)/dᵢ. This gives an exact convergence criterion with no regularity assumption on the support gaps. We contrast it with normal connected locally finite graded domains of Krull dimension two, obtained by adapting an infinite-pole construction of Maclean. Their lower rate is δ/R, where R = lim supᵢ dᵢ₊₁/dᵢ, and their upper rate is δ. In particular, normality, finite dimension and a degree-one nonzerodivisor do not jointly force convergence of the exponential Hilbert rate.

Keywords Euler products · exponential growth · Hilbert functions · graded domains · sparse degree supports

Scope of the work

The Euler-product rates require the stated uniform block-entropy limit and arbitrary increasing degree support. The normal-domain construction adapts Maclean’s infinite-pole method. No age-algebra realization or permutation-class realization is asserted. The manuscript cites prior exponential restricted-partition work; no worldwide priority or verified nonoverlap with every earlier asymptotic result is claimed.

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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.

References and related publications

  1. P. J. Cameron, Sequences realized by oligomorphic permutation groups , J. Integer Seq. 3 (2000), Article 00.1.5. https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html.
  2. B. L. Granovsky and D. Stark, Asymptotic enumeration and logical limit laws for expansive multisets and selections , J. London Math. Soc. (2) 73 (2006), 252–272. arXiv:math/0407322.
  3. B. L. Granovsky and D. Stark, Developments in the Khintchine–Meinardus probabilistic method for asymptotic enumeration , arXiv:1311.2254v2 (2015).
  4. E. Manstavičius and R. Skrabut ̇ enas,On analytic problems for additive arithmetical semigroups , Ann. Univ. Sci. Budapest., Sect. Comp. 22 (2003), 269–285. https://ac.inf.elte.hu/Vol_ 022_2003/doi/269_22.pdf.
  5. M. V. Vakhitov and D. S. Minenkov, On logarithmic asymptotics for the number of restricted partitions in the exponential case , Moscow J. Comb. Number Theory 12 (2023), 297–309. doi:10.2140/moscow.2023.12.297.
  6. C. Maclean, Approximable algebras and a question of H. Chen , arXiv:1703.01801v1 (2017).
  7. C. Maclean, Characterising approximable algebras , arXiv:1709.06945v1 (2017).

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Exponential growth of sparse Euler products and normal graded domains (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/exponential-growth-sparse-euler-products-normal-graded-domains/v/1

BibTeX
@techreport{luminary_exponential_growth_sparse_euler_products_normal_graded_domains_v1,
  title = {Exponential growth of sparse Euler products and normal graded domains},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/exponential-growth-sparse-euler-products-normal-graded-domains/v/1}
}

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