Sierpinski-type spectral self-similar measures

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