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 $p\geq2$ be a positive integer, $E_p^2:=\frac{1}{p}{(i,j)^t:0\leq i,j\leq p-1}$ and $\mathcal{Z}{D}^2:={x\in[0, 1)^2:\sum{d\in D}{e^{2\pi i\langle d,x\rangle}}=0}$. We show that if $\emptyset\neq\mathcal{Z}{D}^2\subset E_p^2\setminus{0}$ 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.