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Na Yuan

Publications and source records attributed to Na Yuan.

3 recordsLinked to original sources
arXiv2025

A winning approach to the intersections of twisted non-recurrent sets with fractals

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set $K\subseteq L$ without assuming any separation condition on $K$. The result is then applied to obtain the Hausdorff dimension of intersections between irreducible self-conformal sets and twisted non-recurrent sets $\mathrm{N}(T,\mathcal{G})$ defined as $$ \mathrm{N}(T,\mathcal{G}):=\left\{\mathbf{x}\in[0,1]^d:\liminf_{n\to\infty}\|T^n(\mathbf{x})-g_n(\mathbf{x})\|>0\right\}, $$ where $T:[0,1]^d\to[0,1]^d$ belongs to a broad class of product maps, $\mathcal{G}:=\{g_n\}_{n\in\mathbb{N}}$ is a sequence of self-maps on $[0,1]^d$ with uniform Lipschitz constant and $\|\cdot\|$ denotes the maximal norm in $\mathbb{R}^d$. When $T$ is the $\beta$-transformation on $[0,1]$, it provides a positive answer to a question raised informally by Broderick, Bugeaud, Fishman, Kleinbock and Weiss (Math. Res. Lett., 2010). For the case $T$ is a $d\times d$ diagonal matrix transformations, our results provide a partial answer asked in a paper of Li, Liao, Velani and Zorin (Adv. Math., 2023). A natural generalization to non-autonomous setting is also obtained.

math.DS↗
arXiv2024

Modified shrinking target problem for Matrix Transformations of Tori

We calculate the Hausdorff dimension of the fractal set \begin{equation*} \Big\{\mathtt{x}\in \mathbb{T}^d: \prod_{1\leq i\leq d}|T_{β_i}^n(x_i)-x_i| < ψ(n) \text{ for infinitely many } n\in \mathbb{N}\Big\}, \end{equation*} where the $T_{β_i}$ is the standard $β_i$-transformation with $β_i>1$, $ψ$ is a positive function on $\mathbb{N}$ and $|\cdot|$ is the usual metric on the torus $\mathbb{T}$. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let $T$ be a $d\times d$ non-singular matrix with real coefficients. Then, $T$ determines a self-map of the $d$-dimensional torus $\mathbb{T}^d:=\mathbb{R}^d / \mathbb{Z}^d$. For any $1\leq i \leq d$, let $ψ_i$ be a positive function on $\mathbb{N}$ and $Ψ(n):=(ψ_1(n),\dots, ψ_d(n))$ with $n\in \mathbb{N}$. We obtain the Hausdorff dimension of the fractal set \begin{equation*} \big\{\mathtt{x}\in \mathbb{T}^d: T^n(x)\in L(f_n(\mathtt{x}), Ψ(n)) \text{ for infinitely many } n\in \mathbb{N}\big\}, \end{equation*} where $L(f_n(\mathtt{x}, Ψ(n)))$ is a hyperrectangle and $\{f_n\}_{n\geq 1}$ is a sequence of Lipschitz vector-valued functions on $\mathbb{T}^d$ with a uniform Lipschitz constant.

math.DS↗