arXiv subjects

Zhi-Yi Wu

Publications and source records attributed to Zhi-Yi Wu.

8 recordsLinked to original sources

arXiv2026

Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture

The eigenvalue problem for a probability measure $μ$ with compact support in $\R$ is whether there exist a countable set $Λ$ and a nonzero real $t\ne 1$ such that both $Λ$ and $tΛ$ are spectra of $μ$, that is, the family $$E_{aΛ}=\{e^{-2πi aλx}:λ\inΛ\}$$ is an orthonormal base for $L^2(μ)$ for $a=1, t$. The eigenvalue problem was discovered independently by Strichartz (Str00), Laba and Wang (LW02) for the Cantor measures $μ_{4,\{0,1\}}$ and $μ_{6,\{0,1,2\}}$, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure $μ_{b,\mathcal{D}}$. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin's primitive root conjecture. To some extent, our results suggest that Artin's primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in (Min57).

math.CA↗

arXiv2026

On the spectrality of the non-homogeneous golden-mean self-similar measure

We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure $μ$, for which the existence of an exponential orthonormal basis in the associated $L^2$-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of $\widehatμ$, which provides concrete evidence that $μ$ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.

math.CA↗

arXiv2026

Almost everywhere convergence of mock Fourier series for the middle-fourth Cantor measure

In 1998, Jorgensen and Pedersen constructed the first example of a singular continuous spectral measure. Precisely, they proved that the self-similar measure generated by the iterated function system $\{\frac{x-1}{4},\frac{x+1}{4}\}$ with equal weights, denoted by $μ_{1/4}$, is a spectral measure with a spectrum $Λ_4$, called the canonical spectrum,\[ Λ_4 := \set{ \sum_{j=0}^{m-1}\varepsilon_j4^j: m\ge 1,\ \varepsilon_j\in\{0,1\} }. \] For $f\in L^1(μ_{1/4})$, let $S_n f$ be the $n$-th partial sum of its Mock Fourier series with respect to $Λ_4$. We prove that the associated maximal operator $S^{\ast}f=\sup_n|S_n f|$ is of weak type $(1,1)$. Consequently, $S_n f\to f$ $μ_{1/4}$-almost everywhere on $\supp(μ_{1/4})$. This solves a long-standing open problem of Strichartz \cite[p.~341]{Str06}.

math.CA↗

arXiv2026

A conditional arithmetic obstruction to the prime numbers as a spectrum of a probability measure

Let $\Pp=\{2,3,5,7,\ldots\}$ denote the set of prime numbers. We prove that if every sufficiently large positive even integer can be represented as a difference of two primes, then there is no Borel probability measure $μ$ on $\R$ for which $\left\{e^{2πi p x}:p\in\Pp\right\}$ is an orthonormal basis of $L^2(μ)$. In particular, under the Polignac conjecture, the prime numbers $\Pp$ cannot be a spectrum (i.e., the set of frequencies of an exponential orthonormal basis) of any probability measure on $\R$.

math.NT↗

arXiv2026

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

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.

math.CA↗

arXiv2025

On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $μ$ is identifying all $t\in\R$ such that both \[\left\{e^{-2πiλx}: λ\inΛ\right\}\quad\text{and}\quad \left\{e^{-2πiλx}: λ\in tΛ\right\}\] form orthonormal bases of the space $L^2(μ)$. Currently, this phenomenon has been observed only in certain singular measures. It is deeply connected to the convergence of Mock Fourier series with respect to the aforementioned bases. In this paper, we apply classical number theory to solve the general conjecture and basic problems in this field within the setting of $N$-Bernoulli convolutions, which extend almost all known results and give some new ones.

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arXiv2023

Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the $s$-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where $s$ is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

math.FA↗

arXiv2022

Spectral measures with arbitrary dimensions

It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of spectral measures. In this paper, we first obtain the formulae of Assouad dimension and of lower dimension for a class of Moran measures in dimension one that is introduced by An and He [J. Funct. Anal. 266 (2014), 343--354]. Based on these results, we show the existence of spectral measures with arbitrary Assound dimensions $\dim_A$ and lower dimensions $\dim_L$ ranging from $0$ to $1$, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures, and prove that the two values may coincide. In fact, more is obtained that for any $0 \leq t \leq s \leq r \leq u\leq 1$, there exists a spectral measure $μ$ such that \[\dim_L μ=t, \dim_H μ=s, \dim_Pμ=r~ \text{and} \dim_Aμ=u,\] where $\dim_H$ and $\dim_P$ denote the Hausdorff dimension and packing dimension of the measure $μ$, respectively. This result improves and generalizes the result of Dai and Sun more simply and flexibly.

math.FA↗