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．Seminars

	European-Asian Joint Webinar on Dynamical Systems

14:00 - 15:00, December 2, 2022 (Friday)
Zoom, Online seminar
(線上演講 Zoom)
Fibers on Well Approximable Set of Systems of Affine Forms
Bo Wang (South China University of Technology)

Abstract

Let $m,n \in \mathbb{N}$ and $\psi: \mathbb{N} \to \mathbb{R}^+$ . Denote by $W_{m.n}(\psi)$ the set of all $(A, \gamma) \in [0,1)^{m \times n} \times [0,1)^m$ such that $\langle A \mathbf{q}-\gamma \rangle < \psi(\|\mathbf{q}\|)$ for infinitely many $\mathbf{q} \in \mathbb{Z}^n$ , where ||• ||denotes supremum norm and $\langle \cdot \rangle$ represents supremum norm distance to $\mathbb{Z}^m$ . Let $\mathcal{D}$ and $\mathcal{C}$ be the sets of all decreasing function $\psi: \mathbb{N} \to \mathbb{R}^+$ satisfying $\sum_{q=1}^{\infty} q^{n-1} \psi(q)^m$ diverges and converges respectively. In this paper, we prove that the union of all the fibers of $\bigcap_{\psi\in \mathcal{D}} W_{m,n}(\psi)$ for $\gamma \in [0,1)^m$ is equal to the complement of badly approximable set. We also give some metrical results of the fibers of $\bigcap_{\psi\in \mathcal{D}} W_{m,n}(\psi)$ for given $A \in [0,1)^{m \times n}$ and $\gamma \in [0,1)^m$ . Our result gives partial answer to Ramírez's questions (2018), he asked whether the fiber of $\bigcap_{\psi\in \mathcal{D}} W_{m,1}(\psi)$ for any given $\gamma \in [0,1)^m$ is non empty and whether the fiber of $\bigcap_{\psi\in \mathcal{D}} W_{m,1}(\psi)$ for some $\gamma \in [0,1)^m$ is of full measure. Then we show that $\bigcup_{\psi\in \mathcal{C}} W_{m,n}(\psi)$ is equal to the complement of the set of badly approximable systems. Moreover, let $(A,\gamma) \in [0,1)^{m\times n} \times [0,1)^m$ , we study the topological structure of the set of all $\psi \in \mathcal{C}$ with $\langle A \mathbf{q} - \gamma \rangle < \psi(\|\mathbf{q}\|)$ for infinitely many $\mathbf{q} \in \mathbb{Z}^n$ .

Link Information: https://us02web.zoom.us/j/88176718266?pwd=MWtnbitsaGZlcDhZNnN2Ym9iSDRVUT09