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Factorization length distribution for affine semigroups V: explicit asymptotic behavior of weighted factorization lengths on numerical semigroups

Stephan Ramon Garcia, Gabe Udell

TL;DR

The paper resolves the problem of describing the asymptotic distribution of weighted factorization lengths in numerical semigroups by a geometric, polytope-based approach. It derives explicit, nonasymptotic error bounds linking discrete factorization counts to Curry–Schoenberg B-spline densities through volumes and lattice-point counts, culminating in an explicit limit distribution for arbitrary linear weights. The central contributions include Theorem A (explicit discrete-to-continuum approximation with an explicit error term), Theorem B (precise bounds comparing vector partition functions with multivariate truncated powers), and Theorem C (explicit bounds for convergence of statistics to the B-spline density), all with fully specified constants. The methods provide a broad framework for studying weighted factorization lengths and potentially other partition-type problems, enabling direct computation of means, variances, and higher moments via the limiting B-spline density and its integrals.

Abstract

We describe the asymptotic behavior of weighted factorization lengths on numerical semigroups. Our approach is geometric as opposed to analytic, explains the presence of Curry-Schoenberg B-splines as limiting distributions, and provides explicit error bounds (no implied constants left unspecified). Along the way, we explicitly bound the difference between the vector partition function and the number of integer points in the variable polytope for a $2 \times k$ matrix.

Factorization length distribution for affine semigroups V: explicit asymptotic behavior of weighted factorization lengths on numerical semigroups

TL;DR

The paper resolves the problem of describing the asymptotic distribution of weighted factorization lengths in numerical semigroups by a geometric, polytope-based approach. It derives explicit, nonasymptotic error bounds linking discrete factorization counts to Curry–Schoenberg B-spline densities through volumes and lattice-point counts, culminating in an explicit limit distribution for arbitrary linear weights. The central contributions include Theorem A (explicit discrete-to-continuum approximation with an explicit error term), Theorem B (precise bounds comparing vector partition functions with multivariate truncated powers), and Theorem C (explicit bounds for convergence of statistics to the B-spline density), all with fully specified constants. The methods provide a broad framework for studying weighted factorization lengths and potentially other partition-type problems, enabling direct computation of means, variances, and higher moments via the limiting B-spline density and its integrals.

Abstract

We describe the asymptotic behavior of weighted factorization lengths on numerical semigroups. Our approach is geometric as opposed to analytic, explains the presence of Curry-Schoenberg B-splines as limiting distributions, and provides explicit error bounds (no implied constants left unspecified). Along the way, we explicitly bound the difference between the vector partition function and the number of integer points in the variable polytope for a matrix.

Paper Structure

This paper contains 17 sections, 20 theorems, 118 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. B-splines
  4. Multivariate truncated powers
  5. Vector partition functions
  6. Lattices
  7. From volume to point counts
  8. Theorem \ref{['Theorem:A']}: weighted factorization length statistics and B-splines
  9. An explicit discrete-volume estimate
  10. Involving the multivariate truncated power
  11. Conclusion of the proof of Theorem \ref{['Theorem:A']}
  12. Theorem \ref{['Theorem:B']}: An explicit bound on the difference between the vector partition function and multivariate truncated power
  13. Theorem \ref{['Theorem:C']}: Bounding convergence of statistics
  14. A technical lemma
  15. Proof of Theorem \ref{['Theorem:C']}
  16. ...and 2 more sections

Key Result

Theorem A

Let $S = \langle n_1,n_2,\ldots,n_k \rangle \subseteq \mathbb{Z}_{\geq 0}$ with If $\mathbf{m}= [m_i] \in \mathbb{R}^k$ and $\mathbf{n} = [n_i] \in \mathbb{Z}_{>0}^k$ are linearly independent, then for all $n\in \mathbb{Z}_{>0}$ and $\alpha,\beta\in \mathbb{R} \cup \{\pm \infty\}$ with $\alpha \leq \beta$, in which $M(t; \frac{m_1}{n_1}, \frac{m_2}{n_2}, \ldots, \frac{m_k}{n_k} )$ is a Curry--

Figures (8)

  • Figure 1: Plots of B-splines for various parameters related to the standard McNugget semigroup $\langle 6,9,20 \rangle$ (the reciprocals of the generators appear as arguments of the B-spline in Theorems \ref{['Theorem:A']} and \ref{['Theorem:C']}).
  • Figure 2: Weighted factorization lengths in the McNugget semigroup $S = \langle 6,9,20 \rangle$ from Example \ref{['Example:Nugget']}. A dot appears above $\ell/n$ at height $n/|\mathsf{Z}_S(n)|$ times the multiplicity of $\ell$ as a weighted factorization length of $n$.
  • Figure 3: Weighted factorization lengths in the non-minimally generated semigroup $S = \langle 2,3,5,8 \rangle$ from Example \ref{['Example:Fibonacci']}. A dot appears above $\ell/n$ at height $n/|\mathsf{Z}_S(n)|$ times the multiplicity of $\ell$ as a weighted factorization length of $n$.
  • Figure 4: Weighted factorization lengths in the non-minimally generated semigroup $S = \langle 2,23,3,47 \rangle$ from Example \ref{['Example:TwoSemis']}. A dot appears above $\ell/n$ at height $n/|\mathsf{Z}_S(n)|$ times the multiplicity of $\ell$ as a weighted factorization length of $n$.
  • Figure 5: Weighted factorization lengths in the numerical semigroup $S = \langle 11,34,35,36 \rangle$ from Example \ref{['Example:Silly']}. A dot appears above $\ell/n$ at height $n/|\mathsf{Z}_S(n)|$ times the multiplicity of $\ell$ as a weighted factorization length of $n$.
  • ...and 3 more figures

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 7
  • Remark 9
  • Remark 13
  • Theorem 14
  • Remark 17
  • Example 18
  • Example 19
  • ...and 44 more