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The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures

Zi-Yun Chen, Zhi-Yi Wu, Min-Min Zhang

TL;DR

The work addresses the Beurling-analytic structure of spectra for a class of self-affine spectral measures, showing that for the plane self-affine measure $\mu_{R,B}$ with $R=2102$ and $B=\{00,10\}$ the spectra can realize any Beurling dimension $t\in[0,1]$ simultaneously with any $t$-Beurling density $s\ge0$. The authors construct spectra $\Lambda_{t,s}$ via explicit $\Lambda_{A,\beta}$-type sets, controlling $\dim_{Be}$ through growth rates of a function $\beta$ and, when needed, adjusting the $t$-Beurling density with additional scaling choices; they also show the level sets of spectra at fixed $t$ are of continuum cardinality. A joint attainment result (Theorem MR3) is achieved by two explicit growth-modulation schemes for $\beta$, covering the cases $s=0$ and $0<s<\infty$. Collectively, the results extend intermediate-value phenomena known for Moran and similar measures to a concrete self-affine spectral measure, illuminating the fine structure of spectral sets in fractal harmonic analysis and suggesting avenues for broader generalizations.

Abstract

It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $μ$, both the Beurling dimension and Beurling density of their spectra attain full flexibility simultaneously. Specifically, for any $t\in (0,\dim_H^w(\supp(μ))]$ and $s\in [0,\infty)$, there exists a spectrum $Λ:=Λ_{t,s}$ of $μ$ satisfying \[\dim_{Be}(Λ)=t\quad\text{and}\quad D_{t}^+(Λ)=s\] where $\dim_H^w$ denotes the pseudo Hausdorff dimension, $\dim_{Be}$ denotes the Beurling dimension and $D_{t}^+$ denotes the $t$-Beurling density. These results provide new insights into the structure of the spectra for a singularly continuous spectral measure.

The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures

TL;DR

The work addresses the Beurling-analytic structure of spectra for a class of self-affine spectral measures, showing that for the plane self-affine measure with and the spectra can realize any Beurling dimension simultaneously with any -Beurling density . The authors construct spectra via explicit -type sets, controlling through growth rates of a function and, when needed, adjusting the -Beurling density with additional scaling choices; they also show the level sets of spectra at fixed are of continuum cardinality. A joint attainment result (Theorem MR3) is achieved by two explicit growth-modulation schemes for , covering the cases and . Collectively, the results extend intermediate-value phenomena known for Moran and similar measures to a concrete self-affine spectral measure, illuminating the fine structure of spectral sets in fractal harmonic analysis and suggesting avenues for broader generalizations.

Abstract

It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures , both the Beurling dimension and Beurling density of their spectra attain full flexibility simultaneously. Specifically, for any and , there exists a spectrum of satisfying where denotes the pseudo Hausdorff dimension, denotes the Beurling dimension and denotes the -Beurling density. These results provide new insights into the structure of the spectra for a singularly continuous spectral measure.
Paper Structure (7 sections, 11 theorems, 44 equations)

This paper contains 7 sections, 11 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectrality and spectra
  4. Beurling density and Beurling dimension
  5. Proofs of Theorems \ref{['MR1']} and \ref{['MR2']}
  6. Proof of Theorem \ref{['MR3']}
  7. Further questions

Key Result

Theorem 1.1

DHLai13DHSW11LW1 For any orthogonal set $\Lambda$ of $\mu_{4,\{0,2\}}$, i.e., $E(\Lambda)$ is orthogonal in $L^2(\mu_{4,\{0,2\}})$, $\dim_{Be}(\Lambda)\leq \frac{1}{2}$, where $\dim_{Be}$ denotes the Beurling dimension. Moreover, for any $t\in [0,\frac{1}{2}]$, there exists a spectrum $\Lambda_t$ su

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • ...and 8 more