The Hausdorff dimension of some exceptional sets of \(p\)-adic integer matrices
A set of linear applications from \(\mathbb{Z}^m\) to \(\mathbb{Z}^n_p\) that very well approximate 0 is studied and its Hausdorff dimension is computed. Namely, for \({\mathbf q}\) in \(\mathbb{Z}^m\), let \(|{\mathbf q} |= \max_{1 \leq i \leq m} |q_i |\) where \(||\) is the ordinary absolute value and, for \({\mathbf q}\) in \(\mathbb{Z}^n_p\), let \(|{\mathbf q} |_p = \max_{1 \leq i \leq n} |{\mathbf q}_i |_p\) where \(||_p\) is the \(p\)-adic absolute value. Let \(X\) denote the set of \(n \times m\) matrices with entries in \(Z_p\). The author studies the set: \[ W (\alpha) = \bigl\{ {\mathbf A} \in X; |{\mathbf q} {\mathbf A} |_p \leq c |{\mathbf q} |^{-\alpha} \text{ for some } c \text{ and infinitely many } {\mathbf q} \in \mathbb{Z}^m \bigr\}. \] The set \(X\) is a compact additive group for its natural topology and can be endowed with its Haar measure \(\eta\). As the dimension of \(X\) is \(nm\), for a ``closed'' ball \(B\) of radius \(r\) one has \(\eta (B) = r^{nm}\). For any subset \(M\) of \(X\) one sets \[ \ell^\gamma (M) = \lim_{r \to 0} \inf_{M \subset \cup B_i} \sum \eta (B_i)^{\gamma/nm} \] where the infimum is taken over all countable coverings \(\{B_i\}\) of \(M\) by balls of radius less than \(r\). The Hausdorff dimension \(\dim (M)\) of \(M\) is, by definition, the unique real such that \(\ell^\gamma (M) = \infty\) for \(\gamma < \dim (M)\) and \(\ell^\gamma (M) = 0\) for \(\gamma > \dim (M)\). For \(m > n\) (more variables than linear forms) and \(\alpha > m/n\), it is proved that: \[ \dim \bigl( W(\alpha) \bigr) = (m - 1) n + m/ \alpha. \] Despite their technically, the proofs are pleasant to read (they contain helpful figures). In the introduction, one finds a useful two page survey about known results in the domain of probabilistic Diophantine approximation.
exceptional sets of \(p\)-adic integer matrices
scientific article; zbMATH DE number 798705