Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

Megumi Asada
Graduate School of Education
Rutgers University
New Brunswick, NJ 08901
USA

Bruce Fang
Department of Mathematics
and Statistics
Williams College
Williamstown, MA 01267
USA

Eva Fourakis
Department of Mathematics
and Statistics
Williams College
Williamstown, MA 01267
USA

Sarah Manski
Center for Statistical Training
and Consulting
Michigan State University
East Lansing, MI 48824
USA

Nathan McNew
Department of Mathematics
Towson University
Towson, MD 21252
USA

Steven J. Miller
Department of Mathematics
and Statistics
Williams College
Williamstown, MA 01267
USA

Gwyneth Moreland
Department of Mathematics, Statistics, and Computer Science
University of Illinois Chicago
Chicago, IL 60607
USA

Ajmain Yamin
Department of Mathematics
CUNY Graduate Center
New York, NY 10016
USA

Sindy Xin Zhang
Department of Mathematics
Tufts University
Medford, MA 02155
USA

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.

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Received June 7 2019; revised versions received October 16 2023; January 20 2024; October 14 2024. Published in Journal of Integer Sequences, October 15 2024.

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