On infinite scalings of the canonical spectrum for self-similar spectral measures

Abstract

Let $(\mu, \Lambda)$ be the canonical spectral pair generated by a Hadamard triple $(N,B,L)$ in $\mathbb{R}$ with $0\in B \cap L$, which means that the family $\big\{ e_\lambda(x)=e^{2\pi \mathrm{i} \lambda x}: \lambda \in \Lambda \big\}$ forms an orthonormal basis in $L^2(\mu)$.We prove that if $\#B < N^{0.677}$, then there are infinitely many primes $p$ such that $(\mu, p\Lambda)$ is also a spectral pair. Under Artin's primitive root conjecture or the Elliott-Halberstam conjecture, the same conclusion holds for $\# B < N$.