Spectrality of one dimensional self-similar measures with consecutive digits

Assume 0<∣ρ∣<10<∣ρ∣<1 and mm is a prime, let μρμρ be the self-similar measure defined by μρ(A)=1m∑j=0m−1μρ(ρ−1A−j), ∀A∈B. We prove that L2(μρ)L2(μρ) contains an orthonormal basis of exponential functions if and only if ρ=±1/mkρ=±1/mk for some k∈Nk∈N.