Scaling of spectra of a class of self‐similar measures on R

Abstract

Let n,b≥2 be two positive integers. For D={0,1,⋯,b−1}, let the self‐similar measure μbn,D be defined by μbn,D=1b∑d∈Dμbn,Dfalse(bnx−dfalse). It is known [18] that μbn,D is a spectral measure with a spectrum Λfalse(bn,Cfalse)=∑j=0finiteajbnj:aj∈C,where C=bn−1false{0,1,⋯,b−1false}. In this paper, we give some conditions on τ∈Z under which the scaling set τΛ(bn,C) is also a spectrum of μbn,D.

Number Theory Problems Related to the Spectrum of Cantor-Type Measures With Consecutive Digits

“…We can compare our results with those of [23,24]. Theorem 1.1 in [23] is the special when p = b n−1 in our Theorem 3.5. Since (pb − 1)/(b − 1) = p + (p − 1)/(b − 1) > p, Theorem 1.2 in [24] is contained in our Corollary 3.6.…”

“…It is easy to see that Z(m (pb) −1 D ) = p{a : a ∈ Z\{0} and b a}. We can compare our results with those of [23,24]. Theorem 1.1 in [23] is the special when p = b n−1 in our Theorem 3.5.…”

“…In this paper, we study the scaling of spectra of a class of self-similar measures with consecutive digits. Wu et al [23,24] considered the same problem and we extend their work by discussing all integers τ.…”