What could the value of the cosmological constant tell us about the
  future variation of the fine structure constant?
  Motivated by reported claims of the measurements of a variation of the fine
structure constant $\alpha$ we consider a theory where the electric charge, and
consequently $\alpha$, is not a constant but depends on the Ricci scalar $R$.
%We then show how this can be considered a particular case of the Bekenstein
theory in which there is no need to %introduce an additional kinetic term for
the scalar field associated to the electric charge, since the Einstein's%
%equations are sufficient to determine the geometry and, consequently the Ricci
scalar. We then study the cosmological implications of this theory, considering
in particular the effects of dark energy and of a cosmological constant on the
evolution of $\alpha$. Some low-red shift expressions for the variation of
$\alpha(z)$ are derived, showing the effects of the equation of state of dark
energy on $\alpha$ and observing how future measurements of the variation of
the fine structure constant could be used to determine indirectly the equation
of state of dark energy and test this theory. In the case of a $\Lambda CDM$
Universe, according to the current estimations of the cosmological parameters,
the present value of the Ricci scalar is $\approx 10%$ smaller than its future
asymptotic value determined by the value of the cosmological constant, setting
also a bound on the future asymptotic value of $\alpha$.
