We calculate the energy band structure for electrons in an external periodic potential combined with a perpendicular magnetic field. Electron-electron interactions are included within a Hartree approximation. The calculated energy spectra display a considerable degree of self-similarity, just as the "Hofstadter butterfly." However, screening affects the butterfly, most importantly the bandwidths oscillate with magnetic field in a characteristic way. We also investigate the dynamic response of the electron system in the far-infrared (FIR) regime. Some of the peaks in the FIR absorption spectra can be interpreted mainly in semiclassical terms, while others originate from inter(sub)band transitions. 73.20.Dx, 73.20.Mf Recent years have witnessed a considerable amount of research effort directed towards understanding of the physics of two-dimensional electron systems (2DES) whose dimensionality is further restricted by man-made periodic potentials and perpendicular magnetic fields. These include quantum dot arrays and antidot superlattices. Concentrating on the latter ones, one distinguishes two principal directions of experimental work: transport studies and far infrared (FIR) spectroscopy. Some of the transport measurements 1,2 have been performed in search of evidence for a self-similar energy spectrum, the so called Hofstadter butterfly. 3 A main theme in the FIR absorption experiments has been to detect and classify the rich variety of collective modes that occur in these systems. [4][5][6] Along with the experimental work, theorists have addressed the same issues. 7-13 The main difficulty lies in the fact that while the superlattice is periodic, the Hamiltonian (including a vector potential) is not. Most recent calculations of superlattice electronic structure have used the Ferrari basis to deal with this matter. 14 We will instead apply ray-group-theoretical techniques 15-18 to effectively reduce the calculational complexity.With this approach, we are able to go beyond earlier calculations. We find the band structure for interacting electrons in general "rational" magnetic fields [i.e., the flux through a unit cell is (L/N )Φ 0 , where Φ 0 is a flux quantum and L, N ∈ Z]. Consequently, we are able to trace even fine-scale features of the butterfly and at the same time study the effects screening has on it. We also explore the FIR response. The resulting spectra are rather rich. Along with absorption peaks caused by collective modes, and known from experiments, 4,5 we find additional ones of mostly quantum-mechanical origin.The antidot superlattice considered here is of simple square symmetry R = n 1 a 1 + n 2 a 2 , with lattice parameter a. The effective one-particle Hamiltonian iswhere the vector potential A = B × r/2 (symmetric gauge) describes the perpendicular magnetic field B and G = g 1 b 1 + g 2 b 2 , denotes the reciprocal lattice vectors. We use GaAs parameters and work with shortperiod superlattices with a = 1000Å and electron density n s = 1.2 · 10 11 cm −2 . A typical magnetic field B = 1.65 T ...