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