Spectrality of a class of infinite convolutions on R

Given an integer m ⩾ 1 . Let Σ ( m ) = { 1 , 2 , … , m } N be a symbolic space, and let { ( b k , D k ) } k = 1 m := { ( b k , { 0 , 1 , … , p k − 1 } t k ) } k = 1 m be a finite sequence pairs, where integers | b k | , p k ⩾ 2 , | t k | ⩾ 1 and p k , t 1 , t 2 , … , t m are pairwise coprime integers for all 1 ⩽ k ⩽ m . In this paper, we show that for any infinite word σ = ( σ n ) n = 1 ∞ ∈ Σ ( m ) , the infinite convolution μ σ = δ b σ 1 − 1 D σ 1 ∗ δ ( b σ 1 b σ 2 ) − 1 D σ 2 ∗ δ ( b σ 1 b σ 2 b σ 3 ) − 1 D σ 3 ∗ ⋯ is a spectral measure if and only if p σ n ∣ b σ n for all n ⩾ 2 and σ ∉ ⋃ l = 1 ∞ ∏ l , where ∏ l = { i 1 i 2 ⋯ i l j ∞ ∈ Σ ( m ) : i l ≠ j , | b j | = p j , | t j | ≠ 1 } .