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

Abstract

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 sets T, where T is generated by an expanding matrix M∈M2(Z) with |det(M)|=4 and D={(0,0)t,(1,0)t,(0,1)t,(−1,−1)t}. We show that T is a spectral set if and only if T is a translational tile. In particular, if T is a spectral set, then Z2 is the unique spectrum of T that contains 0, so it is a tiling set of T by dual criteria.