Spectrality of alternating-sign Moran measures

Abstract

For \(m\geq 2\), let \(D_m=\{0,1,\ldots,m-1\}\). For each \(k\geq 1\), consider the family of contractions \[ \Phi_k=\{\phi_{k,d}:d\in D_{n_k}\}, \qquad \phi_{k,d}(x)=(-1)^d b_k^{-1}(x+d), \] where \(b_k\) and \(n_k\) are integers satisfying \(b_k\geq n_k\geq 2\). We construct the canonical pullback attractor generated by \(\{\Phi_k\}_{k\geq 1}\) and the associated equal-weight Moran measure \(\mu\). Suppose that \(\{b_k\}_{k\geq 1}\) is bounded and that \(n_k\) is even for every \(k\). We prove that \(\mu\) is spectral if \(2\mid b_2,\ n_2\mid 2b_2$, and \(n_k\mid b_k\) for all \(k\geq 3\). If, in addition, \(4\mid n_k\) for every \(k\), then the converse also holds, yielding a complete characterization of spectrality. The proof reduces the associated matrix-valued Fourier recursion to the Fourier transform of an infinite convolution of discrete probability measures. The main new ingredient in the necessity argument is a decomposition method for rational-scale infinite convolutions, in which a spectrum is successively partitioned into congruence classes.