Spectral property of the Bernoulli convolutions

For 0<ρ<1, let μρ be the Bernoulli convolution associated with ρ. Jorgensen and Pedersen [P. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L2-spaces, J. Anal. Math. 75 (1998) 185–228] proved that if ρ=1/q where q is an even integer, then L2(μρ) has an exponential orthonormal basis. We show that for any 0<ρ<1, L2(μρ) contains an infinite orthonormal set of exponential functions if and only if ρ is the nth root of a fraction p/q where p is an odd integer and q is an even integer.