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

Let $(\mu ,\mathrm{\Lambda })$ 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 $\{e_{\lambda }(x)=e^{2\pi \mathrm{i}\lambda x}:\lambda \in \mathrm{\Lambda }\}$ forms an orthonormal basis in $L^2(\mu )$.We prove that if $\mathrm{\#}B<N^{0.677}$, then there are infinitely many primes $p$ such that $(\mu ,p\mathrm{\Lambda })$ is also a spectral pair. Under Artin’s primitive root conjecture or the Elliott-Halberstam conjecture, the same conclusion holds for $\mathrm{\#}B<N$.