Sierpinski-type spectral self-similar measures

We consider the Sierpinski-type self-similar measure μρμρ on R2R2 with contraction ratio 0<|ρ|<10<|ρ|<1, we show that μρμρ is a spectral measure if and only if |ρ|=1/(3p)|ρ|=1/(3p) for some integer p>0p>0. A similar characterization for Bernoulli convolution is due to Dai [2], over which ρ=1/(2p)ρ=1/(2p).