Non-spectral problem for the planar self-affine measures

In this paper, we consider the non-spectral problem for the planar self-affine measures μM,D generated by an expanding integer matrix M∈M2(Z) and a finite digit set D⊂Z2. Let p≥2 be a positive integer, Ep2:=1p{(i,j)t:0≤i,j≤p−1} and ZD2:={x∈[0,1)2:∑d∈De2πi〈d,x〉=0}. We show that if ∅≠ZD2⊂Ep2∖{0} and gcd⁡(det⁡(M),p)=1, then there exist at most p2 mutually orthogonal exponential functions in L2(μM,D). In particular, if p is a prime, then the number p2 is the best.