Badly approximable numbers and vectors in Cantor-like sets (Q2845532)

Let \(d\geq 1\) be an integer. In this paper the authors describe a class of subsets \(\mathcal L\) of \(\mathbb R^d\), and a class \(\mathcal C\) of compact subsets of \(\mathbb R^d\), such that for any \(S \in \mathcal L\) and \(C\in \mathcal C\), \(S \cap C\) is uncountable. The arguments are based on a generalization of Schmidt's game [\textit{W. M. Schmidt}, Trans. Am. Math. Soc. 123, 178--199 (1966; Zbl 0232.10029); Diophantine approximation. York: Springer-Verlag (1980; Zbl 0421.10019)]. For the historic context see also [\textit{C. S. Aravinda} and \textit{E. Leuzinger}, Ergodic Theory Dyn. Syst. 15, No. 5, 813--820 (1995; Zbl 0835.58026); \textit{S. G. Dani}, Comment. Math. Helv. 61, 636--660 (1986; Zbl 0627.22013); Lond. Math. Soc. Lect. Note Ser. 134, 69--86 (1989; Zbl 0705.11042)].NEWLINENEWLINENEWLINENEWLINEDefinitionsNEWLINENEWLINE- A real number \(t\) is said to be \textit{badly approximable} if there exists \(\delta > 0\) such that for every rational number \(\frac{p}{q}\) we have \(|t- \frac{p}{q}|\geq \frac{\delta^2}{q^2}\).NEWLINENEWLINE- Let \(\mathbb R^d, d \geq 1\), equipped with the usual \textit{Hilbert norm} and the corresponding metric denoted by \(\|\cdot\|\) and \(d(\cdot,\cdot)\) respectively. For subsets \(S_1\) and \(S_2\) of \(\mathbb R^d\), \(d(S_1,S_2)\) denotes \(\inf\{\|x-y\|\;| \;x \in S_1, y \in S_2\}\). For \(x \in \mathbb R^d\) and a subset \(S\), \(d(x,S)\) stands for \(d(\{x\}, S)\). For a subset \(S\) of \(\mathbb R^d\) the \textit{thickness} of \(S\) is defined by \(\tau (S) = \inf_V\sup_{x,y\in S} d(x-y,V)\), where the infimum is taken over all hyperplanes \(V\) in \(\mathbb R^d\).NEWLINENEWLINE- Let \(C\) be a compact subset of \(\mathbb R^d, d \geq 1\). A \textit{filter of coverings} of \(C\) is a sequence \(C_0, C_1,\dots,\) where each \(C_n\) is a family of compact subsets covering \(C\), such that the two following conditions are satisfied: (i) \(C_0\) = \{C\}, and for \(m \geq n \geq 0\) every \(Y \in C_m\) is contained in some \(X\) in \(C_n\). (ii) For all \(n \geq 0\) every \(X \in C_n\) is covered by the sets \(Y\) from \(C_{n+1}\) that are contained in \(X\).NEWLINENEWLINE- Let \(\{C_n\}\) be a filter of coverings on a compact subset \(C\) of \(\mathbb R^d\). It is said to be a \textit{fine filter of coverings} (FFC for short) if the two following conditions are satisfied: (i) there exist \(\theta\in (0, 1)\) and \(a, b > 0\) such that for any \(n = 0, 1, \dots\) and \(X \in C_n\), \(a\theta^n\leq\) diam\((X) \leq b\theta^n\). (ii) there exists \(\sigma\in (0, \frac{1}{2})\) such that for any even integer \(n \geq 0\) and \(X \in C_n\), if \(Y \in C_{n+1}\) and \(Y \subset X\), then diam\((Y ) \leq \sigma\tau(X)\).NEWLINENEWLINE- Let the torus \(\mathbb T^d\) realized as \(\mathbb R^d/\mathbb Z^d\) and \(\pi: \mathbb R^d\rightarrow \mathbb T^d\) denote the canonical map. Let \(F\) be the subgroup of \(\mathbb T^d\) consisting of all elements of finite order, namely \(F = \pi(\mathbb Q^d)\), which is a dense subgroup of \(\mathbb T^d\).NEWLINENEWLINE{Results:} The three following theorems are characteristic of the kind of all the results obtained by the authors in this article:NEWLINENEWLINETheorem. Let \(C\) be a compact subset of \(\mathbb R\) admitting an \(FFC\). For each \(i \in \mathbb N\), let \(\Omega_i\) be a neighbourhood of \(C\) in \(\mathbb R\) and \(f_i : \Omega_i \rightarrow \mathbb R\) be a \textit{bi-Lipschitz map}. Then there exist uncountably many \(t\) in \(C\) such that \(f_i(t)\) is badly approximable for all \(i\).NEWLINENEWLINETheorem. Let \(C\) be a compact subset of \(\mathbb R^d, d\geq 2\), admitting an FFC. Let \(\{A_i\}\) be a sequence of \textit{affine automorphisms} of \(\mathbb R^d\) into itself. Then there exist uncountably many \(v\) in C such that \(A_i(v)\) is badly approximable for all \(i\).NEWLINENEWLINETheorem. Let \(C\) be a compact subset of \(\mathbb R^d, d \geq 1\), admitting an FFC. Then there exists an uncountable subset \(E\) of \(C\) such that for any \(v \in E\) and any surjective endomorphism \(\rho\) of \(\mathbb T^d\), \(C_\rho(\pi(v)) \cap F = \emptyset\).