Number theoretic considerations related to the scaling spectrum of self-similar measures with consecutive digits

Abstract

Let p,b≥2 be two integers, and let D=p{0,1,…,b−1}. It is well known that the self-similar measure μpb,D generated by the iterated function system {τd(x)=(pb)−1(x+d)}d∈D is a spectral measure with a spectrum Λ(pb,C)=∑j=0finite(pb)jcj:cj∈C={0,1,…,b−1}.In this paper, we study a class of primitive and composite numbers and their related properties. And we also explore the simple prime numbers and the properties related to their order. Based on these, let r be a positive integer, we give some conditions on the distinct prime numbers t1,t2,…,tr such that the scaling set ∏i=1rtikiΛ(pb,C) is also a spectrum of μpb,D for all ki≥0.