Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

TL;DR

This paper investigates the problem of avoiding 3-term geometric progressions in the non-commutative setting of Hurwitz quaternions, providing exact formulas and bounds for the density of such progression-free sets.

Contribution

It extends the study of progression-free sets to Hurwitz quaternions, a non-commutative ring, and derives explicit density formulas and bounds.

Findings

Exact density formula for progression-free sets in Hurwitz quaternions

Upper and lower bounds for the supremum of densities

Analysis of non-commutativity effects on the problem

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.