Fourier bases on general self-similar Sierpinski measures

Abstract

Let |ρ|>1 be a real number and let Q be a 2 × 2 orthogonal and involutory matrix. Let R={0,u,v}⊂Z2 such that det(u,v)≠0 and 3∤det(u,v). In this paper, we prove that the Hilbert space L2(μρQ,R) generated by the general self-similar Sierpinski measure μρQ,R has an orthonormal basis of the form EΛ=e2πi〈λ,x〉:λ∈Λwith Λ⊂R2, i.e., μρQ,R is a spectral measure, if and only if there exist three integers d,m,n such that d2∣(m2+n2) and ρQ=3md3nd3nd−3md in the standard expression. This provides new and different characterization of spectrality of self-similar measures in the plane and gives a counterexample to Łaba–Wang’s conjecture in two dimension.