Non-spectral problem for the planar self-affine measures

Abstract

In this paper, we consider the non-spectral problem for the planar self-affine measures ${\mu }_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset {\mathbb{Z}}^2$. Let $p\ge 2$ be a positive integer, $E_p^2:=\frac{1}{p}\{(i,j{)}^t:0\le i,j\le p-1\}$ and ${\mathcal{Z}}_D^2:=\{x\in [0,1{)}^2:{\sum }_{d\in D}{e}^{2\pi i⟨d,x⟩}=0\}$. We show that if $\varnothing \ne {\mathcal{Z}}_D^2\subset E_p^2\setminus \{0\}$ and $\gcd (\det (M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2({\mu }_{M,D})$. In particular, if $p$ is a prime, then the number $p^2$ is the best.

14 pages

References in corpus (1)

• Hadamard triples generate self-affine spectral measures