Spectrality of generalized Sierpinski-type self-affine measures

Abstract

For an expanding integer matrix M ∈ M 2 (Z) and an integer digit setIn [5,36], the authors separately investigated the spectral property of the measure µ M,D in the case of det(M) 3Z or α 1 β 2 − α 2 β 1 3Z. In this paper, we consider the remaining case where det(M) ∈ 3Z and α 1 β 2 − α 2 β 1 ∈ 3Z, and give the necessary and sufficient conditions for µ M,D to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure µ M,D .