On spectral N-Bernoulli measures

For 0<ρ<1 and N>1 an integer, let μ be the self-similar measure defined by μ(⋅)=∑i=0N−11Nμ(ρ−1(⋅)−i). We prove that L2(μ) has an exponential orthonormal basis if and only if ρ=1q for some q>0 and N divides q. The special case is the Cantor measure with ρ=12k and N=2 [16], which was proved recently to be the only spectral measure among the Bernoulli convolutions with 0<ρ<1 [4].