The cardinality of certain -orthogonal exponentials

The self-affine measures μ M , D corresponding to the case (i) M = p I 3 , D = { 0 , e 1 , e 2 , e 3 } in the space R 3 and the case (ii) M = p I 2 , D = { 0 , e 1 , e 2 , e 1 + e 2 } in the plane R 2 are non-spectral, where p > 1 is odd, I n is the n × n identity matrix, and e 1 , … , e n are the standard basis of unit column vectors in R n . One of the non-spectral problem on μ M , D is to estimate the number of orthogonal exponentials in L 2 ( μ M , 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 L 2 ( μ M , D ) each, and the number 4 is the best.