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 L(μρ) has an exponential orthonormal basis. We show that for any 0 < ρ < 1, L(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. © 2008 Elsevier Inc. All rights reserved. MSC: primary 42C05, 42A65; secondary 28A78, 28A80