arXiv subjects

Chun-Kit Lai

Publications and source records attributed to Chun-Kit Lai.

At least 19 recordsLinked to original sources

arXiv2026

Spectrality of factors of product spectral measures

We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting. Furthermore, we establish a new class of spectral unions of intervals for which the product spectral set question has a positive answer. More precisely, if $A$ is a subset of the natural numbers such that $A\oplus B = \{0,1,\cdots, N-1\}$ for some $B\subset \mathbb N$ and $N>1$ then the product measure $\mathcal{L}|_{A+[0,1]}\times \nu$ is a spectral measure (that may be singular) if and only if $\nu$ is a spectral measure.

math.CA↗

arXiv2026

Hausdorff dimension of images and graphs of some random complex series

Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random complex series $S(x)=\sum_{n=1}^{\infty}a_n X_n\phi_n(\lambda_nx)$, where $\lambda_n$ is an increasing sequence with $\sup_n\lambda_{n+1}/\lambda_n<\infty$ and $\phi_n$ satisfies some uniform Lipschitz and boundedness conditions. This class of series includes the famous Weierstrass and Riemann functions as well as others appeared in literature. These results help predict the exact values of the deterministic cases.

math.CA↗

arXiv2026

On Constructions of full-dimensional absolutely normal sets of uniqueness

We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{\H{o}}s, and LeVeque \cite{DEL1963} and properties of the order of integers in the multiplicative groups. The construction of these measures differs from the class of pointwise absolutely normal self-similar measures introduced by Hochman and Shmerkin \cite{Hochman2015}, in which dynamical approaches were used. As an application, for all gauge functions $\varphi(r)$ with $r/\varphi(r)\to 0$ as $r\to 0$, we obtain a set of uniqueness $K$ with ${\mathcal H}^{\varphi}(K)>0$. Moreover, we show that there exists a pointwise absolutely normal measure $ \mu $ of dimension one fully supported on $K$. The result demonstrates that having a lot of absolutely normal numbers in a Cantor set, even with dimension one, cannot guarantee that it supports a measure with Fourier decay. It also shows that the ${\mathsf{DEL}}$ criterion being satisfied for all integers does not guarantee any Fourier decay nor the supporting set is a set of multiplicity.

math.CA↗

arXiv2025

Distinct dimensions for attractors of bi-Lipschitz iterated function systems

In this paper, we construct an iterated function system on the line consisting of two bi-Lipschitz contractions whose attractor has distinct lower, Hausdorff, lower box, upper box, and Assouad dimensions, thereby providing negative answers to certain folklore questions. Furthermore, as a by-product of our study of bi-Lipschitz IFSs, we construct IFSs within this family that exhibit interesting fractal behaviour. In particular, we prove the following two statements: (i) There exists a bi-Lipschitz IFS for which the pushforward of any ergodic measure with positive entropy is not exact dimensional; (ii) There exists a bi-Lipschitz IFS whose attractor has empty interior yet positive Lebesgue measure.

math.DS↗

arXiv2025

Non-spectrality of some piecewise smooth curves and unions of line segments

We develop a systematic study about the spectrality of measures supported on piecewise smooth curves by studying the support of the tempered distributions arising from the tiling equation of some singular spectral measures. In doing so, we show that the arc-length measures of all closed polygonal lines are not spectral. {In particular, the boundary of a square is not spectral. We also show that the ``plus space'' (two crossing line segments) is not spectral.} Furthermore, our theory also shows that the arc length measures on {smooth} convex curves with finitely many transverse self-intersections are not spectral. Finally, several natural open questions about the spectrality of singular measures and {piecewise} smooth curves will also be discussed.

math.CA↗

arXiv2025

A non-sticky Kakeya set of Lebesgue measure zero

The Kakeya set conjecture in ${\mathbb R} ^3$ was recently resolved by Wang and Zahl. The distinction between sticky and non-sticky Kakeya sets plays an important role in their proof. Although the proof did not require the Kakeya set to be Lebesgue measure zero, measure zero Kakeya sets are the crucial case whose study is required to resolve the conjecture. In this paper, we explicitly construct a non-sticky Kakeya set of Lebesgue measure zero in ${\mathbb R}^2$ (and hence in any dimension). We also construct non-trivial sticky and non-sticky Kakeya sets in high dimension that are not formed by taking the Cartesian product of a 2-dimensional Kakeya set with ${\mathbb R}^{d-2}$, and we verify that both Kakeya sets have Hausdorff dimension $d$.

math.CA↗

arXiv2025

Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$

We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Fa\`a di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $\alpha$-stable process has Fourier dimension 1 almost surely when $\alpha \in [1,2]$ and is a Salem set when $\alpha = 1$.

math.PR↗

arXiv2025

When is the fractal uncertainty principle for discrete Cantor sets most uncertain?

We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of the spectral pair studied in the spectral sets literature. We investigate distributed spectral pairs in some cyclic groups and some complete classifications are given. Finally, we also discuss the most uncertain case in the continuous setting.

math.CA↗

arXiv2024

Fifty years of the Erd\H{o}s similarity conjecture

Erd\H{o}s similarity conjecture was proposed by P. Erd\H{o}s in 1974. The conjecture remains open for exponentially decaying sequences as well as Cantor sets that have both Newhouse thickness and Hausdorff dimension zero. In this article, written after 50 years of the conjecture being proposed, we review progress on some new variants of the original problem: namely, the bi-Lipschitz variant, the topological variant, and a variant ``in the large''. These problems were recently studied by the authors and their collaborators. Each of them offers new perspectives on the original conjecture.

math.CA↗

arXiv2024

Interior of certain sums and continuous images of very thin Cantor sets

We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the interior is robust under small perturbation of the mapping. More generally, we can also show that the image set $H(\alpha, K_1,K_2)$, where $H$ is some $C^1$ function on ${\mathbb R}^N\times{\mathbb R}^d\times{\mathbb R}^d$ with non-vanishing Jacobian, have non-empty interior for $\alpha$ all in an open ball of ${\mathbb R}^N$. This result allows us to show that all Cantor sets are not topologically universal using $C^1$ local diffeomorphism, proving a stronger version of the topological Erd\H{o}s similarity conjecture. Moreover, we are also able to construct a Cantor set of dimension $d$ on ${\mathbb R}^{2d}$, whose distance set has an interior.

math.MG↗

arXiv2024

Topological Erd\H{o}s similarity conjecture and strong measure zero sets

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a result of the fact that the Borel conjecture is independent of the \textsf{ZFC} axiomatic set theory, the existence of an uncountable topologically universal set is independent of the \textsf{ZFC}. Moreover, our results can also be generalized to locally compact Polish groups ${\mathbb G}$. Returning to the measure side, we pose Full-Measure universal Erd\H{o}s Similarity Conjecture with strongly meager sets via the duality of measure and category.

math.CA↗

arXiv2023

Erd\H{o}s similarity problem via bi-Lipschitz embedding

The Erd\H{o}s similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences. In this paper, we relax the problem to the bi-Lipschitz embedding and obtain some sharp criteria about the bi-Lipschitz Erd\H{o}s similarity problem for strictly decreasing sequences.

math.CA↗

arXiv2023

Riesz bases of exponentials for multi-tiling measures

Let $G$ be a closed subgroup of ${\mathbb R}^d$ and let $ν$ be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group $G^{\perp}$. We form a multi-tiling measure $μ= μ_1+...+μ_N$ where $μ_i$ is translationally equivalent to $ν$ and different $μ_i$ and $μ_j$ have essentially disjoint support. We obtain some necessary and sufficient conditions for $μ$ to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of $G = {\mathbb Z}\times {\mathbb R}$ and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups ${\mathbb Z}\times \{0\}$. This rules out a natural candidate of potential Riesz basis for the square boundary.

math.FA↗

arXiv2023

Projections of totally disconnected thin fractals with very thick shadows on ${\mathbb R}^d$

We study an extreme scenario of the Mastrand projection theorem for which a fractal has the property that its orthogonal projection is the same as the orthogonal projection of its convex hull. We extend results in current literature and establish checkable criteria for self-affine sets to have such property. Using this, we show that every convex polytope on $\R^d$ contains a totally disconnected compact set, which is a union of self-affine sets, of dimension as close to 1 as possible, as well as a rectifiable 1-set, such that the fractal projects to an interval in every 1-dimensional subspace and its convex hull is the given polytope. Other convex sets and projections onto higher dimensional subspaces will also be discussed.

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

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↗