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

Abstract

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