Spectrality of one dimensional self-similar measures with consecutive digits

Assume 0 < ∣ ρ ∣ < 1 and m is a prime, let μ ρ be the self-similar measure defined by μ ρ ( A ) = 1 m ∑ j = 0 m − 1 μ ρ ( ρ − 1 A − j ) , ∀ A ∈ B . We prove that L 2 ( μ ρ ) contains an orthonormal basis of exponential functions if and only if ρ = ± 1 / m k for some k ∈ N .