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

TL;DR

The paper connects self-similar spectral measures generated by Hadamard triples to number-theoretic properties of primes. It shows that when the digit set is small relative to the base, there are infinitely many prime spectral eigenvalues for the canonical spectrum, and under strong conjectures these results hold in broader regimes. A key technical bridge is the equivalence condition for spectral eigenvalues expressed via intersections of rational Cantor-set points with modular lattices, coupled with order-growth properties of primes. The work highlights deep interactions between fractal spectral theory and analytic number theory, offering both unconditional and conditional results with explicit quantitative statements. The outcomes advance understanding of scaling phenomena in spectral measures and suggest pathways for leveraging prime distribution conjectures to classify spectral eigenvalues.

Abstract

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

Theorems & Definitions (24)

• Conjecture 3
• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 2.1
• Theorem 2.2
• Proposition 2.3