Baker-Schmidt theorem for Hausdorff dimensions in finite characteristic
For any $n \in \mathbb{N}$ and any $\lambda \in \mathbb{R}_{\geq1}$, let $\mathcal{K}_n(\lambda)$ denote the set of all $\zeta \in \mathbb{R}$ such that, for any real $\lambda'< \lambda$, there exist infinitely many algebraic $\alpha$ of degree at most $n$ satisfying $|\zeta - \alpha| < H(\alpha)^{-(n+1)\lambda'}$, where $H(\alpha)$ denotes the (usual) height. Moreover, let $\mathcal{K}_n'(\lambda)$ be the subset in $\mathcal{K}_n(\lambda)$ consisting of elements not belonging to $\mathcal{K}_n(\tau)$ for all $\tau > \lambda$. Then \textit{A. Baker} and \textit{W. M. Schmidt} [Proc. Lond. Math. Soc. (3) 21, 1--11 (1970; Zbl 0206.05801)] studied the Hausdorff dimensions of these sets and obtained the following result: For any $n$ and $\lambda$ as above, the Hausdorff dimensions of $\mathcal{K}_n(\lambda)$ and $\mathcal{K}_n'(\lambda)$ both equal to $\frac{1}{\lambda}$. Whereas the translations of the Baker-Schmidt theorem from the above-mentioned real case to the complex or $p$-adic case are quite straightforward, the situation for fields of power series over finite fields treated in the present paper is rather different. One new difficulty arises, e.g., from the separability of an irreducible polynomial over the rationals, which is used to claim the nonvanishing of the discriminant.
scientific article; zbMATH DE number 6882146
scientific article