I am attending Rene Pfister’s talk at the Shanghai Institute of Mathematics and Interdisciplinary Sciences in Shanghai. This is joint work with N. de Saxcé

\(K\)-rational Approximation to Linear Subspaces

For every \(x \in \mathbb{R}\setminus \mathbb{Q}\), there exist infinitely many pairs \((p,q) \in \mathbb{Z}\times \mathbb{N}\), with \(p\) and \(q\) coprime, such that \[\begin{equation} \left| x - \frac{p}{q} \right| < \frac{1}{q^{2}}. \end{equation}\]

Let \(K\) be an algebraic number field, and set \(n = [K : \mathbb{Q}]\). Let \(1 \le k, \ell < d\). Given an \(\ell\)-dimensional linear subspace \(X \subseteq \mathbb{R}^{d}\), find a \(k\)-dimensional linear subspace \(v\) of \(\mathbb{R}^{d}\), defined over \(K\), that is close to \(X\).

A subspace \(v \subseteq \mathbb{R}^{d}\) is defined over \(K\) if \[\begin{equation} \mathop{\mathrm{span}}_{\mathbb{R}}\bigl(v \cap K^{d}\bigr) = v. \end{equation}\]

Diophantine setup

Write \[\begin{equation} X_{\ell}(\mathbb{R}) = \mathrm{Gr}_{\ell, d}(\mathbb{R}), \qquad X_{k}(K) = \{\, k\text{-dimensional subspaces of } K^{d}\,\}. \end{equation}\]

Distance

For \(x \in X_{\ell}(\mathbb{R})\) and \(y \in X_{k}(\mathbb{R})\) define \[\begin{equation} d(x, y) \;\coloneqq\; \max_{\substack{y' \subseteq y \\ \dim y' = 1}} \min_{\substack{x' \subseteq x \\ \dim x' = 1}} \frac{\bigl|\vec{x}\wedge\vec{y}\bigr|}{\|\vec{x}\|\,\|\vec{y}\|}, \end{equation}\] where \(\vec{x}\) spans \(x'\) (1-dim), \(\vec{y}\) spans \(y'\) (1-dim); \(x\) is \(\ell\)-dim and \(y\) is \(k\)-dim.

Height

Over \(\mathbb{Q}\)

For \(v \in X_{k}(\mathbb{Q})\), \[\begin{equation} H_{\mathbb{Q}}(v) = \operatorname{covol}\bigl(v \cap \mathbb{Z}^{d} \text{ in } v\bigr) = \|v_{1} \wedge \cdots \wedge v_{k}\|, \end{equation}\] where \(\|\cdot\|\) is the norm on \(\bigwedge^{k}\mathbb{R}^{d}\) and \(v_{1},\dots,v_{k}\) is a \(\mathbb{Z}\)-basis of \(v\cap\mathbb{Z}^{d}\). Equivalently, \[\begin{equation} H_{\mathbb{Q}}(v) \;=\; \prod_{s \in M_{\mathbb{Q}}} \|v_{1} \wedge \cdots \wedge v_{k}\|_{s}. \end{equation}\]

Over \(K\)

One defines similarly \[\begin{equation} H_{K}(v) \;=\; \prod_{w \in M_{K}} \|v_{1} \wedge \cdots \wedge v_{k}\|_{w}. \end{equation}\]

Diophantine exponents

Main theorem

For all \(H \ge 1\), \[\begin{equation} \bigl|\{\, v \in X_{k}(\mathbb{Q}) \,:\, H_{\mathbb{Q}}(v) < H \,\}\bigr| \;\asymp\; H^{d}. \end{equation}\]

For \(v \in X_{k}(K)\), look at \[\begin{equation} M_{v} \;=\; \bigl\{\, x \in X_{\ell}(\mathbb{R}) \,:\, v \subseteq x \,\bigr\} \;\subseteq\; X_{\ell}(\mathbb{R}). \end{equation}\] The set \(\{\, x : d(x, v) \le \gamma \,\}\) is an \(r\)-tubular neighbourhood of \(M_v\) in \(X_\ell(\mathbb{R})\), and it has volume \[\begin{equation} \;\asymp\; \gamma^{\dim M_{v}} \;=\; \gamma^{k(d-\ell)}. \end{equation}\] So the total measure of \(x\)’s that get \(\gamma\)-approximated by some \(v\) with \(H(v)\le H\) is \[\begin{equation} \;\asymp\; H^{d}\,\gamma^{k(d-\ell)}. \end{equation}\] Choosing \(\gamma \asymp H^{-\beta}\) gives \[\begin{equation} 1 \;\asymp\; H^{d} \cdot H^{-\beta\,k(d-\ell)}. \end{equation}\] It is therefore desirable to have \[\begin{equation} \beta \;\ge\; \frac{d}{k(d-\ell)} \end{equation}\] in order to have shrinking targets. This gives the almost-sure value.

Comparison of heights

For \(v \in X_{k}(\mathbb{Q}) \subseteq X_{k}(K)\), \[\begin{equation} H_{K}(v) \;=\; H_{\mathbb{Q}}(v)^{[K:\mathbb{Q}]}. \end{equation}\]

Sketch of the proof

(Step 1)

For all \(x \in X_{\ell}(\mathbb{R})\), [ _{k,K}(x) ;=; , ] where \(\gamma_{k,K}(x)\) is the .

(Step 2)

Use ergodicity of the diagonal action to show that the escape rate vanishes almost surely.

Proof of Step 1 (case \(k = \ell = 1\) in dimension \(d\))

We aim for \[\begin{equation} \beta_{K}(x) \;\le\; \frac{d}{d - 1 - \gamma_{K}(x)}. \end{equation}\] Let \[\begin{equation} a(t) \;=\; \operatorname{diag}\bigl(e^{-(d-1)t},\, e^{t},\, \dots,\, e^{t}\bigr) \;\in\; \mathrm{SL}_{n}(\mathbb{R}) \;\subseteq\; \mathrm{SL}_{n}(\mathbb{A}), \end{equation}\] and consider the projection \[\begin{equation} P : \mathrm{SL}_{n}(\mathbb{R}) \longrightarrow X_{\ell}(\mathbb{R}), \qquad g \longmapsto \langle g \cdot e_{1}, \dots, e_{\ell} \rangle, \qquad P_{g} \longmapsto g^{-1} x_{0}. \end{equation}\]

Goal: A relation between the Diophantine exponent of \(x = P_{u}(x)\) and the asymptotic behaviour of \(a(t)\, u(x)\, K^{d} \subseteq \mathbb{A}^{d}\): \[\begin{equation} \gamma_{K}(x) \;=\; \liminf_{t \to \infty}\, \frac{-1}{t}\, \log \lambda\bigl(a(t)\, u(x)\, K^{d}\bigr), \end{equation}\] where \[\begin{equation} \lambda(g K^{d}) \;=\; \inf\Bigl\{\, \|g v\| \,:\, v \in K^{d} \setminus \{0\} \,\Bigr\} \;=\; \prod_{s} \|(g v)_{s}\|_{s}. \end{equation}\]