Spectrality of the planar Sierpinski family

Let μ   be a Borel probability measure with compact support in R2R2. μ   is called a spectral measure if there exists a countable set Λ⊂R2Λ⊂R2 such that EΛ={e−2πi〈λ,x〉:λ∈Λ}EΛ={e−2πi〈λ,x〉:λ∈Λ} is an orthonormal basis for L2(μ)L2(μ). In this note we prove that the integral Sierpinski measure μA,D is a spectral measure if and only if (A,D) is admissible. This completely settles the spectrality of integral Sierpinski measures in R2R2.