On zeros and spectral property of self-affine measures

Abstract

Abstract In this paper, we study the spectral property of self-affine measures μ M , D generated by an expanding integer matrix M ∈ M n ( Z ) and a finite digit set D ⊂ Z n , i.e. investigate whether the function in L 2 ( μ M , D ) has a Fourier expansion. Let Z D n = { x ∈ [ 0 , 1 ) n : ∑ d ∈ D e 2 π i ⟨ d , x ⟩ = 0 } and E q n = ( q − 1 Z n ∩ [ 0 , 1 ) n ) ∖ { 0 } for some integer q ⩾ 2 . Through the discussion of the relationships between Z D n and E q n , we establish some criteria to determine whether μ M , D is spectral or non-spectral. Indeed, under those suitable assumptions, if μ M , D is a spectral measure, we find its spectrum; otherwise, we give the maximum number of orthogonal exponential functions in L 2 ( μ M , D ) . As an application, our results can contain some well-known conclusions.