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Electronic spectrum of the three-dimensional quasi crystal

The electronic spectrum of icosahedral quasicrystal with central atom decoration of Amman-Mackay network is investigated in the tight-binding approximation. Quasicrystal is described as a structural limit of the optimal cubic approximants with increasing period. The electronic spectra for the first four optimal cubic approximants do not contain hierarchical gap structure which is typical for Cantor set of the spectrum of one-dimensional quasicrystal. At the same time the spectrum with increasing the order of approximant becomes singular on the whole energy scales.