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

TL;DR

This paper investigates the conditions under which self-similar spectral measures maintain their spectral properties when scaled by infinitely many primes, revealing new thresholds related to the size of the digit set.

Contribution

It establishes that for certain bounds on the digit set size, infinitely many prime scalings preserve the spectral measure property, under specific number-theoretic conjectures.

Findings

Infinitely many primes p make (μ, pΛ) spectral if #B < N^{0.677}.

Under Artin's and Elliott-Halberstam conjectures, the threshold extends to #B < N.

The results connect spectral measures with deep number-theoretic conjectures.

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 $\# B < N$.