Spectrality of digit sets and spectral self-affine measures

Abstract

The present paper determines the spectrality of digit set $D$ relating to a spectral self-affine measure $\mu_{M,D}$. This is motivated by a conjecture of Dutkay, Han and Jorgensen. The conjecture states that $D$ is always a spectral set if $\mu_{M,D}$ is a spectral measure in the dimension $n=1$. For a self-affine measure $\mu_{M,D}$, we obtain several conditions for the digit set $D$ to be a spectral set. The result here provides some supportive evidence for the conjecture. It also generalizes the corresponding known result in a certain case.