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Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, Sindy Xin Zhang

Abstract

Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields and polynomial rings over finite fields. We study the analogous problem in the Hurwitz quaternion order to see how non-commutativity affects the problem. We compute an exact formula for the density of a 3-term geometric-progression-free set of Hurwitz quaternions arising from a greedy algorithm and derive upper and lower bounds for the supremum of upper densities of 3-term geometric-progression-free sets of Hurwitz quaternions.

Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

Abstract

Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields and polynomial rings over finite fields. We study the analogous problem in the Hurwitz quaternion order to see how non-commutativity affects the problem. We compute an exact formula for the density of a 3-term geometric-progression-free set of Hurwitz quaternions arising from a greedy algorithm and derive upper and lower bounds for the supremum of upper densities of 3-term geometric-progression-free sets of Hurwitz quaternions.

Paper Structure

This paper contains 9 sections, 14 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Combinatorics of Hurwitz quaternions
  3. Exact densities of sets of Hurwitz quaternions
  4. Bounds on the supremum of upper densities
  5. Lower bound
  6. Upper bound
  7. The quaternion greedy set
  8. Future work
  9. Acknowledgments

Key Result

Lemma 1

The number of Hurwitz quaternions of norm $N$ is $24\cdot \sigma_{\emph{odd}}(N)$, where $\sigma_{\emph{odd}}$ is the sum-of-odd-divisors function

Theorems & Definitions (32)

  • Lemma 1
  • Lemma 2
  • proof : Proof of (1)
  • proof : Proof of (2)
  • Definition 3
  • Example 4
  • Theorem 5
  • proof
  • Example 6
  • Theorem 7
  • ...and 22 more