  In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic
length spectra are necessarily commensurable. While this is known to be true
for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open.
Building towards a negative answer to this question, Futer and Millichap
recently constructed infinitely many pairs of non-commensurable, non-arithmetic
hyperbolic 3-manifolds which have the same volume and whose length spectra
begin with the same first $m$ geodesic lengths. In the present paper, we show
that this phenomenon is surprisingly common in the arithmetic setting. In
particular, given any arithmetic hyperbolic 3-orbifold derived from a
quaternion algebra, any finite subset $S$ of its geodesic length spectrum, and
any $k \geq 2$, we produce infinitely many $k$-tuples of arithmetic hyperbolic
3-orbifolds which are pairwise non-commensurable, have geodesic length spectra
containing $S$, and have volumes lying in an interval of (universally) bounded
length. The main technical ingredient in our proof is a bounded gaps result for
prime ideals in number fields lying in Chebotarev sets which extends recent
work of Thorner.
