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Greedy Regular Convolutions

Jan Snellman

TL;DR

Greedy Regular Convolutions study a spectrum of regular, homogeneous, and bounded convolutions on arithmetical functions, extending unitary and ternary cases by greedily packing integers into at most $d$-element blocks. The method constructs $\Pi^{(d)}$ and $\mathcal{S}^{(d)}$ via the trees $T_d(n)$ and the matrix $A^{(d,N)}$, and analyzes primitive elements and their ranks. The paper proves uniqueness for the unitary and ternary cases among equal-block-size convolutions, and provides a detailed account for $d=3$ (including the primitive structure and densities) and explorations for $d=4$, with selective sifting as a key tool. It also raises several conjectures about higher block-lengths, natural densities, and heights, illustrating a rich combinatorial structure with connections to known integer sequences.

Abstract

We introduce a class of convolutions on arithmetical functions that are regular in the sense of of Narkiewicz, homogeneous in the sense of Burnett et al, and bounded, in the sense that there exists a common finite bound for the rank of primitive numbers. Among these "greedy convolutions" the unitary convolution and the "ternary convolution" are particularly interesting: they are the only regular, homogeneous convolutions where each primitive number have the same finite rank. While the greedy convolution of length 3, also described in detail, has primitive numbers of rank 3 and rank 1, it is still special in that the set of primitives can be generated by a simple recursive procedure that we name selective sifting.

Greedy Regular Convolutions

TL;DR

Greedy Regular Convolutions study a spectrum of regular, homogeneous, and bounded convolutions on arithmetical functions, extending unitary and ternary cases by greedily packing integers into at most -element blocks. The method constructs and via the trees and the matrix , and analyzes primitive elements and their ranks. The paper proves uniqueness for the unitary and ternary cases among equal-block-size convolutions, and provides a detailed account for (including the primitive structure and densities) and explorations for , with selective sifting as a key tool. It also raises several conjectures about higher block-lengths, natural densities, and heights, illustrating a rich combinatorial structure with connections to known integer sequences.

Abstract

We introduce a class of convolutions on arithmetical functions that are regular in the sense of of Narkiewicz, homogeneous in the sense of Burnett et al, and bounded, in the sense that there exists a common finite bound for the rank of primitive numbers. Among these "greedy convolutions" the unitary convolution and the "ternary convolution" are particularly interesting: they are the only regular, homogeneous convolutions where each primitive number have the same finite rank. While the greedy convolution of length 3, also described in detail, has primitive numbers of rank 3 and rank 1, it is still special in that the set of primitives can be generated by a simple recursive procedure that we name selective sifting.

Paper Structure

This paper contains 18 sections, 18 theorems, 93 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Greedy convolutions
  3. Convolution products on arithmetical functions
  4. Homogeneous bounded regular convolutions
  5. Unitary convolution
  6. Ternary convolution
  7. Greedy convolutions
  8. Uniqueness of the unitary and the ternary convolution
  9. The greedy convolution of length 3
  10. Primitive elements
  11. Statistics of the greedy convolution of length 3
  12. Selective sifting
  13. Questions and conjectures regarding selective sifting
  14. The greedy convolution of length 4
  15. Questions about greedy convolutions
  16. ...and 3 more sections

Key Result

Theorem 1

The convolution $*_{A}$ is regular if and only if $A$ is simple, reflexive, symmetric, transitive, and multiplicative, and for each prime $p$ and each positive $k$, and moreover $p^{t} \in A(p^{2t})$, $p^{2t} \in A(p^{3t})$, and so on. Equivalently, let $K$ denote the class of all decompositions of ${\mathbb{N}}$ into arithmetical progressions (finite or not) containing zero, such that no two pro

Figures (9)

  • Figure 1: Distance between consecutive primitives for the ternary convolution
  • Figure 2: The tree $T_{1}(9)$
  • Figure 3: The tree $T_{2}(15)$
  • Figure 4: The tree $T_{3}(40)$
  • Figure 5: The tree $T_{4}(40)$
  • ...and 4 more figures

Theorems & Definitions (43)

  • Theorem 1: Narkiewicz
  • Definition 2
  • Lemma 3
  • Lemma 4
  • proof
  • Corollary 5
  • Corollary 6
  • proof
  • Definition 7
  • Lemma 8
  • ...and 33 more