Baker-Schmidt theorem for Hausdorff dimensions in finite characteristic
type: Item
instance of: scholarly article
title: Baker-Schmidt theorem for Hausdorff dimensions in finite characteristic
zbMATH Open document ID: 1443.11133
author: Huei-Jeng Chen
published in: Finite Fields and their Applications
publication date: 2018-06-07T00:00:00Z
review text: 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.
reviewed by: Peter Bundschuh
Mathematics Subject Classification ID: 11J17
Mathematics Subject Classification ID: 11J61
zbMATH DE Number: 6882146
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: finite characteristic
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: j.ffa.2018.04.012
OpenAlex ID: W2806376290
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Wikidata QID: Q129733620
DOI: 10.1016/J.FFA.2018.04.012
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description: scientific article; zbMATH DE number 6882146