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Exponential growth of sparse Euler products and normal graded domains

Scott Schubr

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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.

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Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-24 · public

Research details

Subjects
Mathematics · Algebra · Combinatorics
Keywords
Euler products, exponential growth, Hilbert functions, graded domains, sparse degree supports
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

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. The manuscript cites prior exponential restricted-partition work.

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References and related resources

  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).