  We investigate the dimensionally induced phase transition from the normal to
the Bose-Einstein-condensed phase for a weakly interacting Bose gas in an
optical lattice. To this end we make use of the Hartree-Fock-Bogoliubov-Popov
theory, where we include numerically exact hopping energies and effective
interaction strengths. At first we determine the critical chemical potential,
where we find a much better agreement with recent experimental data than a pure
Hartree-Fock treatment. This finding is in agreement with the dominant role of
quantum fluctuations in lower dimensions, as they are explicitly included in
our theory. Furthermore, we determine for the 1D-3D-transition the power-law
exponent of the critical temperature for two different non-interacting Bose gas
models yielding the same value of 1/2, which indicates that they belong to the
same universality class. For the weakly interacting Bose gas we find for both
models that this exponent is robust with respect to finite interaction
strengths.
