Non-spectral problem of self-affine measures with consecutive collinear digits in \(\mathbb{R}^2\)

Let \(\mu_{M,D}\) ${\mu }_{M,D}$ be the planar self-affine measure generated by an expanding integer matrix \(M\in M_2(\mathbb{Z})\) $M\in M_2(\mathbb{Z})$ and an integer digit set \(D=\{0,1,\dots,q-1\}v\) $D=\{0,1,\cdots ,q-1\}v$ with \(v\in\mathbb{Z}^2\setminus\{0\}\) $v\in {\mathbb{Z}}^2\backslash \{0\}$ , where \(\gcd(\det(M),q)=1\) $gcd(det(M),q)=1$ and \(q\ge 2\) $q\ge 2$ is an integer. If the characteristic polynomial of \(M\) $M$ is \(f(x)=x^2+\det(M)\) $f(x)=x^2+det(M)$ and \(\{v, Mv\}\) $\{v,Mv\}$ is linearly independent, we show that there exist at most \(q^2\) $q^2$ mutually orthogonal exponential functions in \(L^2(\mu_{M,D})\) $L^2({\mu }_{M,D})$ , and the number \(q^2\) $q^2$ is the best. In particular, we further give a complete description for the case \(M= {\rm diag}(s, t)\) $M=\mathrm{diag}(s,t)$ with \(\gcd(st, q)=1\) $gcd(st,q)=1$ . This extends the results of Wei and Zhang [24].