Spectral and tiling properties for a class of planar self-affine sets

It is well known that Fuglede's conjecture gives a connection between the spectrality and geometrical tiling property. In this paper, we consider a class of planar self-affine set , where is generated by an expanding matrix & ISIN; 2(Z) with | det()| = 4 and D = {(0 ,0) , (1 , 0) , (0 ,1) , (-1 , -1)}. We show that is a spectral set if and only if is a translational tile. In particular, if is a spectral set, then Z2 is the unique spectrum of that contains 0 , so it is a tiling set of by dual criteria.