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Lingmin Liao

Publications and source records attributed to Lingmin Liao.

At least 19 recordsLinked to original sources
arXiv2026

Hausdorff dimension of $\tau$-approximable points on self-similar sets in $\mathbb R^d$

Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}| 0$ such that, for every $1/d<\tau<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(\tau)}(K\cap W_d(\tau))=\infty, \qquad\text{with } s(\tau):=\delta+\frac{d+1}{1+\tau}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(\tau)) = \delta+\frac{d+1}{1+\tau}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud--Durand conjectural formula for $\tau>1$ sufficiently close to $1$.

math.NT↗
arXiv2026

Jarn\'ik-type theorem for self-similar sets

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote the set of $(\psi_\tau,\boldsymbol{\theta})$-well approximable numbers, where $\psi_\tau(q)=q^{-\tau}$. Assuming a local estimate for the $\mu$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbol{\theta})/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(\psi_\tau,\boldsymbol{\theta})$, together with some nontrivial lower bounds when $\tau$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $\theta=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(\psi_\tau,0)$ has full $(\delta+\frac{2}{1+\tau}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in $\mathbb R$, affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and B\'enard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset[0,1]^d$ whose intersection with $W_d(\psi_\tau,0)$ has full $(\delta+\frac{1+d}{1+\tau}-d)$-Hausdorff measure.

math.NT↗
arXiv2024

Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup

We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring $\limsup$ sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for $τ>0$, let $S(τ)$ denote the set of $(x_1,x_2)\in \mathbb{R}^2$ simultaneously satisfying the inequalities $\|q x_1 \| \, < \, q^{-τ} $ and $ \|q x_2 \| \, < \, e^{-q}$ for infinitely many $q \in \mathbb{N}$. Then, the `unbounded' Mass Transference Principle enables us to show that $\dim_{\rm H} S(τ) \, = \, \min \big\{ 1, 3/(1+τ) \big\} \, $.

math.NT↗
arXiv2024

Diophantine approximation by rational numbers of certain parity types

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

math.NT↗
arXiv2024

Julia sets and geometrically finite maps over finite extensions of the $p$-adic field

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $ϕ\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $ϕ$ is the natural restriction of $\mathbb{C}_p$-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that $ϕ$ is geometrically finite, we prove that the dynamics on the $K$-Julia set of $ϕ$ is a countable state Markov shift.

math.DS↗
arXiv2024

Expanding property and statistical laws for $p$-adic subhyperbolic rational maps

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $ϕ\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $ϕ$ is eventually periodic. We show that subhyperbolic maps in $K(z)$ exhibit expanding property with respect to some (singular) metric. As an application, under a mild assumption, we establish several statistical laws for such maps in $K(z)$ with compact $\mathbb{C}_p$-Julia sets.

math.DS↗
arXiv2023

Quantitative recurrence properties for piecewise expanding maps on $ [0,1]^d $

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ μ$. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\}) $ the set of points $ \mathbf x $ such that $ T^n\mathbf x\in \mathbf x+H_n $ for infinitely many $ n\in\mathbb N $, where $ \mathbf x+H_n $ is the translation of $ H_n $. We prove that if $ μ$ is exponential mixing and the density of $ μ$ is sufficiently regular, then the $μ$-measure of $ \mathcal R(\{H_n\}) $ is zero or full according to the sum of the volumes of $ H_n $ converges or not. In the case that $ T $ is a matrix transformation, our results extend a previous work of Kirsebom, Kunde, and Persson [to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., 2023] in two aspects: by allowing the matrix to be non-integer and by allowing the `target' sets $ H_n $ to be hyperrectangles or hyperboloids. We also obtain a dimension result when $ T $ is a diagonal matrix transformation.

math.DS↗
arXiv2023

The Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problem

Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a self-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a large class of subsets (namely, those satisfying the so called bounded property $ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids) we show that the $d$-dimensional Lebesgue measure of the shrinking target set $W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{ 1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n \})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the third-named author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.

math.NT↗
arXiv2022

Path-dependent shrinking targets in generic affine iterated function systems

We calculate the Hausdorff dimension of shrinking target sets of symbolic balls in generic affine iterated function systems with matrix norms bounded by $\tfrac 12$. Relative to earlier works of e.g. Barany and Troscheit, and Koivusalo and Ramirez, we impose only very mild condition, known as complete reducibility, on the iterated function system. We also generalise these earlier works in that the size of the target in the current work is allowed to depend on the trajectory hitting it. The formula for Hausdorff dimension we obtain is presented as a zero point of certain pressure function.

math.DS↗
arXiv2022

Uniform approximation problems of expanding Markov maps

Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ μ_ϕ$ be the invariant Gibbs measure associated with a Hölder continuous potential $ ϕ$. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y| 0 $ and $ x\in[0,1] $. The critical value of $ κ$ such that $ \textrm{dim}_{\textrm H}\mathcal U^κ(x)=1 $ for $ μ_ϕ$-a.e.$ \, x $ is proven to be $ 1/α_{\max} $, where $ α_{\max}=-\int ϕ\,dμ_{\max}/\int\log|T'|\,dμ_{\max} $ and $ μ_{\max} $ is the Gibbs measure associated with the potential $ -\log|T'| $. Moreover, when $ κ>1/α_{\max} $, we show that for $ μ_ϕ$-a.e.$ \, x $, the Hausdorff dimension of $ \mathcal U^κ(x) $ agrees with the multifractal spectrum of $ μ_ϕ$.

math.DS↗
arXiv2021

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.

math.NT↗
arXiv2021

Uniform random covering problems

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $ω=(ω_n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set \[ \mathcal U (ω):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ \| ω_n -y \| < r_N \}. \] Some sufficient conditions for $\mathcal U(ω)$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that $\mathcal U(ω)$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

math.PR↗
arXiv2020

Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

math.DS↗