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Chun-Kit Lai

Publications and source records attributed to Chun-Kit Lai.

At least 37 records · Page 2Linked to original sources
arXiv2022

On a topological Erdős similarity problem

A pattern is called universal in another collection of sets, when every set in the collection contains some linear and translated copy of the original pattern. Paul Erdős proposed a conjecture that no infinite set is universal in the collection of sets with positive measure. This paper explores an analogous problem in the topological setting. Instead of sets with positive measure, we investigate the collection of dense $G_δ$ sets and in the collection of generic sets (dense $G_δ$ and complement has Lebesgue measure zero). We refer to such pattern as topologically universal and generically universal respectively. It is easy to show that any countable set is topologically universal, while any set containing an interior cannot be topologically universal. In this paper, we will show that Cantor sets on ${\mathbb R}^d$ are not topologically universal and Cantor sets with positive Newhouse thickness on ${\mathbb R}^1$ are not generically universal. This gives a positive partial answer to a question by Svetic concerning the Erdős similarity problem on Cantor sets. Moreover, we also obtain a higher dimensional generalization of the generic universality problem.

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↗
arXiv2022

Hausdorff and Fourier dimension of graph of continuous additive processes

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive additive processes with zero mean. Such processes can be represented as $X_t = B_{V(t)}$ where $B$ is Brownian motion and $V$ is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if $V$ is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if $V$ admits at least one point with positive lower Hölder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the $L^q$ spectrum of $V$. We will show that if $V$ is generated by a self-similar measure on ${\mathbb R}^{1}$ with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its $L^q$ spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is $1+\frac12\cdot\frac{\log 2}{\log 3}$. Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely.

math.PR↗
arXiv2022

Large sets avoiding affine copies of infinite sequences

A conjecture of Erdős states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = \{2^{-k} : k \geq 1\}$. In this article, we consider infinite decreasing sequences $A = \{a_k: k \geq 1\}$ in ${\mathbb R}$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1}) = e^{φ(n)} $, where $φ(n)/n\to 0$ as $n\to\infty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Borel set ${\mathcal O}\subseteq \mathbb R$ of Hausdorff dimension 1, but Lebesgue measure zero, that avoids all nontrivial affine copies of $A\cup\{0\}$.

math.CA↗
arXiv2020

On exponential bases and frames with non-linear phase functions and some applications

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(Λ,φ) = \{e^{2πi λ\cdotφ(x)}: λ\inΛ\}$ where the phase function $φ$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $(Λ,φ)$ for which $E(Λ,φ)$ is an orthogonal basis or a frame for $L^{2}(μ)$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $μ$ is the Lebesgue measure on $[0,1]$ and $Λ= {\mathbb{Z}},$ we show that only the standard phase functions $φ(x) = \pm x$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $φ$ defined on ${\mathbb{R}}^{d}$ such that the system $E(Λ,φ)$ is an orthonormal basis for $L^{2}[0,1]^{d}$ when $d\geq2.$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.

math.FA↗
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↗
arXiv2020

Spectral properties of some unions of linear spaces

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $ρ$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(ρ)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the "L" shape at the origin, which has a unique orthonormal basis up to translations of the form \[ \left\{e^{2 πi (λ_1 x_1 + λ_2 x_2)} : (λ_1, λ_2) \in Λ\right\}, \] where \[ Λ= \{ (n/2, -n/2) \mid n \in {\mathbb Z} \}. \]

math.FA↗
arXiv2020

Spectrum is rational in dimension one

A bounded measurable set $Ω\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\{e^{2πi \langleλ,x\rangle}: λ\inΛ\}$ for $L^2(Ω)$. In this paper, we show that in dimension one $d=1$, any spectrum $Λ$ with $0\inΛ$ of a spectral set $Ω$ with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on ${\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\mathbb Z}_{n}.$

math.FA↗
arXiv2020

A proof of the Erdös similarity conjecture

We show that for any infinite set $A$ in ${\mathbb R}$, there exists a compact set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erdös similarity conjecture.

math.CA↗
arXiv2019

Conjugate Phase Retrieval in Paley-Wiener Space

We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_π$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling both $f$ and $f^{\prime}$ only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

cs.IT↗
arXiv2019

Existence and exactness of exponential Riesz sequences and frames for fractal measures

We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.

math.FA↗
arXiv2019

Conjugate Phase Retrieval on ${\mathbb C}^M$ by real vectors

In this paper, we will introduce the notion of {\it conjugate phase retrieval}, which is a relaxed definition of phase retrieval allowing recovery of signals up to conjugacy as well as a global phase factor. It is known that frames of real vectors are never phase retrievable on ${\mathbb C}^M$ in the ordinary sense, but we show that they can be conjugate phase retrievable in complex vector spaces. We continue to develop the theory on conjugate phase retrievable real frames. In particular, a complete characterization of conjugate phase retrievable real frames on ${\mathbb C}^2$ and ${\mathbb C}^3$ is given. Furthermore, we show that a generic real frame with at least $4M - 6$ measurements is conjugate phase retrievable in ${\mathbb C}^M$ for $ M \ge 4.$

math.FA↗
arXiv2019

Fourier frames for surface-carried measures

In this paper we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the first example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups.

math.CA↗
arXiv2019

Self-affine spectral measures and frame spectral measures on ${\mathbb R}^d$

We study Fourier bases on invariant measures generated by affine iterated function systems in ${\mathbb R}^d$ with integer coefficients. We show that, for simple digit sets, these systems satisfy the open set condition and have no overlap. We present natural geometric conditions under which such measures have an orthonormal basis or a frame of exponential functions with frequencies being a subset of ${\mathbb Z}^d$. Moreover, we characterize when such measures have a spectrum in ${\Bbb Z}^d$.

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↗
arXiv2018

Undersampled windowed exponentials and their applications

We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form $$ {\mathcal F}(g): =\{e^{2πi n x}: n\ge 0\}\cup \{g(x)e^{2πi nx}: n<0\} $$ for $L^2[-1/2,1/2]$ by the spectra of the Toeplitz operators with symbol $g$. Using this characterization, we classify all real-valued functions $g$ such that ${\mathcal F}(g)$ is complete or forms a frame/basis. Conversely, we use the classical Kadec-1/4-theorem in non-harmonic Fourier series to determine all $ξ$ such that the Toeplitz operators with symbol $e^{2πi ξx}$ is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, as an application, we use our results to answer some open questions in dynamical sampling, phase retrieval and derivative samplings on $\ell^2({\mathbb Z})$ and Paley-Wiener spaces of bandlimited functions.

math.FA↗
arXiv2018

Non-separable lattices, Gabor orthonormal bases and Tilings

Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $Λ=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(Λ)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}χ_K, Λ)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block triangular matrix with diagonal matrices $A$ and $B$, we prove that if $\mathcal G(|K|^{-1/2}χ_K, Λ)$ is an orthonormal basis, then $K$ can be written as a finite union of fundamental domains of $A({\mathbb Z}^d)$ and at the same time, as a finite union of fundamental domains of $B^{-t}({\mathbb Z}^d)$. If $A^tB$ is an integer matrix, then there is only one common fundamental domain, which means $K$ tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domains is also possible. We also provide a constructive way for forming a Gabor window functions for a given upper triangular lattice. Our study is related to a Fuglede's type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

math.FA↗