The cardinality of certain μM,D-orthogonal exponentials

The self-affine measures μM,D corresponding to the case (i) M=pI3, D={0,e1,e2,e3} in the space R3 and the case (ii) M=pI2, D={0,e1,e2,e1+e2} in the plane R2 are non-spectral, where p>1 is odd, In is the n×n identity matrix, and e1,…,en are the standard basis of unit column vectors in Rn. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D) and to find them. In the present paper we show that, in both cases (i) and (ii), there are at most 4 mutually orthogonal exponentials in L2(μM,D) each, and the number 4 is the best.