  Motivated by the open problem of exhibiting a subset of Euclidean space which
has no exponential Riesz basis, we focus on exponential Riesz bases in finite
abelian groups. We point out that that every subset of a finite abelian group
has such a basis, removing interest in the existence question in this context.
We then define tightness quantities for subsets to measure the conditioning of
Riesz bases; for normalized tightness quantities, a value of one corresponds to
an orthogonal basis, and a value of infinity corresponds to nonexistence of a
basis. As an application, we obtain new weak evidence in favor of the open
problem by giving a sequence of subsets of finite abelian groups whose
tightness quantities go to infinity in the limit. We also prove that the
Cartesian product of a set with a finite abelian group has the same tightness
quantities as the original set. Lastly, under an additional hypothesis,
explicit bounds are given for tightness quantities in terms of a subset's
lowest multi-tiling level by a subgroup and its geometric configuration. This
establishes a quantitative link between discrete geometry and harmonic analysis
in this setting.
