  We present a detailed beyond-mean-field analysis of a weakly interacting Bose
gas in the crossover from three to low dimensions. We find an analytical
solution for the energy and provide a clear qualitative picture of the
crossover in the case of a box potential with periodic boundary conditions. We
show that the leading contribution of the confinement-induced resonance is of
beyond-mean-field order and calculate the leading corrections in the three- and
low-dimensional limits. We also characterize the crossover for harmonic
potentials in a model system with particularly chosen short- and long-range
interactions and show the limitations of the local-density approximation. Our
analysis is applicable to Bose-Bose mixtures and gives a starting point for
developing the beyond-mean-field theory in inhomogeneous systems with
long-range interactions such as dipolar particles or Rydberg-dressed atoms.
