Scaling of spectra of self-similar measures with consecutive digits

Abstract

We consider the equally-weighted Cantor measures μp,q generated by the iterated function system (IFS) {fi(x)=xp+iq}i=0q−1, where 2≤q∈Z and q<p∈R. It is known that if q divides p, then μp,q is a spectral measure with a spectrumΛp,q={0,1,…,q−1}+p{0,1,…,q−1}+p2{0,1,…,q−1}+⋯(finite sum) (Dai, He and Lai (2013) [6]). In this paper the authors study some positive integers b such that bΛp,q is also a spectrum of μp,q.