Spectrality of a class of infinite convolutions on R *

Abstract

Given an integer m⩾1 . Let Σ(m)={1,2,…,m}N be a symbolic space, and let {(bk,Dk)}k=1m:={(bk,{0,1,…,pk−1}tk)}k=1m be a finite sequence pairs, where integers |bk| , pk⩾2 , |tk|⩾1 and pk,t1,t2,…,tm are pairwise coprime integers for all 1⩽k⩽m . In this paper, we show that for any infinite word σ=(σn)n=1∞∈Σ(m) , the infinite convolution μσ=δbσ1−1Dσ1∗δ(bσ1bσ2)−1Dσ2∗δ(bσ1bσ2bσ3)−1Dσ3∗⋯ is a spectral measure if and only if pσn∣bσn for all n⩾2 and σ∉⋃l=1∞∏l , where ∏l={i1i2⋯ilj∞∈Σ(m):il≠j,|bj|=pj,|tj|≠1} .