Spectral property of self-affine measures on ${\mathbb R}^n$

TL;DR

This paper investigates the spectral properties of self-affine measures generated by expanding integer matrices and specific digit sets, providing conditions for spectrality and orthogonality, extending one-dimensional results to higher dimensions.

Contribution

It offers new sufficient and necessary conditions for the spectrality of self-affine measures in higher dimensions, generalizing previous one-dimensional findings.

Findings

Identifies conditions for spectral measures and orthogonal exponentials

Provides necessary and sufficient criteria in special cases

Extends one-dimensional spectral measure results to higher dimensions

Abstract

We study spectral properties of the self-affine measure $\mu_{M,\mathcal {D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in \mathbb{Z}^n\setminus\{0\}$ and $q\ge 2$ is an integer. Some sufficient conditions for $\mu_{M,\mathcal {D}}$ to be a spectral measure or to have infinitely many orthogonal exponentials are given. Moreover, for some special cases, we can obtain a necessary and sufficient condition on the spectrality of $\mu_{M,\mathcal {D}}$. Our study generalizes the one dimensional results proved by Dai, {\it et al.} (\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}).