  We propose a lower limit on the size of a single discrete gravitational extra
dimension in the context of an effective field theory for massive gravitons.
The limit arises in this setup from the requirement that the Casimir energy
density of quantum fields is in agreement with the observed dark energy density
of the universe. The Casimir energy densities can be exponentially suppressed
to an almost arbitrarily small value by the masses of heavy bulk fields,
thereby allowing a tiny size of the extra dimension. This suppression is only
restricted by the strong coupling scale of the theory, which is known to be
related to the compactification scale via an UV/IR connection for local
gravitational theory spaces. We thus obtain an upper limit on the
compactification scale of the discrete gravitational extra dimension in the
range 10^7 GeV...10^12 GeV, while the strong coupling scale is by a factor 10^2
larger than the compactification scale. We also comment on a possible
cancelation of the gravitational contribution to the quantum effective
potential.
