Orthogonal exponentials on the ball

We show that there does not exist an infinite sequence of vectors λn in d, d > 1, such that the corresponding exponentials ei λn,x , x d, when considered on the unit ball B in d, are pairwise orthogonal in L2(B) (B being endowed with Lebesgue measure). The weaker result that L2(B) does not have an infinite orthogonal base of exponentials has recently been established by A. Iosevich, N. Katz, and S. Pedersen in [2]. For d = 2 the present result was announced in the authorʹs 1974 paper [1].