Orthogonal exponential functions of self-similar measures with consecutive digits in R

Suppose that 0<|ρ|<1 and m≥2 is an integer. Let μρ,m be the self-similar measure defined by μρ,m(⋅)=1m∑j=0m−1μρ,m(ρ−1(⋅)−j). In this paper, we prove that L2(μρ,m) contains an infinite orthonormal set of exponential functions if and only if ρ=±(q/p)1/r for some p,q,r∈N+ with gcd(p,q)=1 and gcd(p,m)>1.