Spectrality of infinite convolutions with three-element digit sets

Abstract

Let 0 < ρ< 1 and let an,bn n=1∞ be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure μρ, 0,an,bn generated by the following infinite convolution product μρ,0,an,bn=δρ 0,a1,b1∗δρ2 0,a2,b2∗δρ3 0,a3,b3∗⋯in the weak convergence, where δE=1E∑e∈Eδe and gcd= 1 for all n∈ N. In this paper, we show that L2(μρ,0,an,bn) admits an exponential orthonormal basis if and only if ρ∈ 3 N and an,bn≡1,2 mod3 for all n∈ N.