arXiv subjects

Li-Xiang An

Publications and source records attributed to Li-Xiang An.

5 recordsLinked to original sources

arXiv2026

Characterization on spectra of self-similar measures and the dual spectral set conjecture

A discrete set $\Lambda$ is called a {\it spectrum} of a Borel probability measure $\mu$ if the exponential functions $\{e^{2\pi i \langle\lambda, x\rangle}:\lambda\in\Lambda\}$ form an orthonormal basis for $L^2(\mu)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by {\L}aba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.

math.CA↗

arXiv2023

Product-form Hadamard triples and its spectral self-similar measures

In a previous work by Łaba and Wang, it was proved that whenever there is a Hadamard triple $(N,{\mathcal D},{\mathcal L})$, then the associated one-dimensional self-similar measure $μ_{N,{\mathcal D}}$ generated by maps $N^{-1}(x+d)$ with $d\in{\mathcal D}$, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples $(N, {\mathcal A}_k, {\mathcal L}_k)$ by putting each triple into different scales of $N$. Our main result is to prove that the associated self-similar measure $μ_{N,{\mathcal D}}$ is a spectral measure. This result allows us to show that product-form self-similar tiles are spectral sets as long as the tiles in the group ${\mathbb Z}_N$ obey the Coven-Meyerowitz $(T1)$, $(T2)$ tiling condition. Moreover, we show that all self-similar tiles with $N = p^αq$ are spectral sets, answering a question by Fu, He and Lau in 2015. Finally, our results allow us to offer new singular spectral measures not generated by a single Hadamard triple. Such new examples allow us to classify all spectral self-similar measures generated by four equi-contraction maps, which will appear in a forthcoming paper.

math.CA↗

arXiv2022

Classification of spectral self-similar measures with four-digit elements

Let $μ$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<ρ<1$. We study when $μ$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2πi λx}\}_{λ\inΛ}$ in $L^2(μ)$. By combining previous results of many authors and a careful study of some new cases, we completely classify all spectral self-similar measures with four maps. Moreover, the case allows us to propose a modified Łaba-Wang conjecture concerning when the self-similar measures are spectral in general cases.

math.CA↗

arXiv2020

Arbitrarily sparse spectra for self-affine spectral measures

Given an expansive matrix $R\in M_d({\mathbb Z})$ and a finite set of digit $B$ taken from $ {\mathbb Z}^d/R({\mathbb Z}^d)$. It was shown previously that if we can find an $L$ such that $(R,B,L)$ forms a Hadamard triple, then the associated fractal self-affine measure generated by $(R,B)$ admits an exponential orthonormal basis of certain frequency set $Λ$, and hence it is termed as a spectral measure. In this paper, we show that if #$B<|\det (R)|$, not only it is spectral, we can also construct arbitrarily sparse spectrum $Λ$ in the sense that its Beurling dimension is zero.

math.FA↗

arXiv2019

On Spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem

In this paper, we show that if we have a sequence of Hadamard triples $\{(N_n,B_n,L_n)\}$ with $B_n\subset \{0,1,..,N_n-1\}$ for $n=1,2,...$, except an extreme case, then the associated Cantor-Moran measure $$ \begin{aligned} μ= μ(N_n,B_n) =& δ_{\frac{1}{N_1}B_1}\astδ_{\frac{1}{N_1N_2}B_2}\ast δ_{\frac{1}{N_1N_2N_3}B_3}\ast...\\ =& μ_n\astμ_{>n} \end{aligned} $$ with support inside $[0,1]$ always admits an exponential orthonormal basis $E(Λ) = \{e^{2πi λx}:λ\inΛ\}$ for $L^2(μ)$, where $Λ$ is obtained from suitably modifying $L_n$. Here, $μ_n$ is the convolution of the first $n$ Dirac measures and $μ_{>n}$ denotes the tail-term. We show that the completeness of $E(Λ)$ in general depends on the ``equi-positivity" of the sequence of the pull-backed tail of the Cantor-Moran measure $ν_{>n}(\cdot) = μ_{>n}((N_1...N_n)^{-1}(\cdot))$. Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of $\{ν_{>n}\}$. This result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research, whose focus has been specifically on $\#B_n\le 4$. Using the Bourgain's example that a sum of sine can be asymptotically small, we shows that, in the extreme case, there exists some Cantor-Moran measure such that the equi-positive condition fails and the Fourier transform of the associated $ν_{>n}$ uniformly converges on some unbounded set.

math.CA↗