Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions
Combinatorics
2024-01-23 v2 Number Theory
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.
Keywords
Cite
@article{arxiv.1807.07605,
title = {Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions},
author = {Megumi Asada and Bruce Fang and Eva Fourakis and Sarah Manski and Nathan McNew and Steven J. Miller and Gwyneth Moreland and Ajmain Yamin and Sindy Xin Zhang},
journal= {arXiv preprint arXiv:1807.07605},
year = {2024}
}
Comments
Version 3.0, 18 pages