  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.
