Higher Diophantine approximation exponents and continued fraction symmetries for function fields. II
type: Item
instance of: scholarly article
zbMATH Open document ID: 1272.11127
DOI: 10.1090/S0002-9939-2013-11538-8
author: Dinesh S. Thakur
published in: Proceedings of the American Mathematical Society
publication date: 2013-07-05T00:00:00Z
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11J93
zbMATH DE Number: 6184459
zbMATH Keywords: function fields
zbMATH Keywords: diophantine approximation exponents
zbMATH Keywords: continued fractions
MaRDI profile type: Publication
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full work available at URL: s0002-9939-2013-11538-8
OpenAlex ID: W2060153247
title: Higher Diophantine approximation exponents and continued fraction symmetries for function fields. II
review text: Definitions: For a non-zero algebraic number \(\beta\), define \(H(\beta)\) to be the maximum of the absolute values of the coefficients of a non-trivial irreducible polynomial with co-prime integral coefficients that it satisfies. For \(\alpha\) an irrational real number not algebraic of degree \(\leq d\), define \(E_d(\alpha)\), (\(E_{\leq d}(\alpha)\) respectively) as \(\lim\sup(-\log |\alpha-\beta|/\log H(\beta)),\) where \(\beta\) varies through all algebraic real numbers of degree \(d\) (\(\leq d\) respectively). Note that \(E_1(\alpha)\) is the usual exponent \(E(\alpha):=\lim\sup (-\log|\alpha-P/Q|/\log |Q|)\). For irrational \(\alpha\), we have \(E(\alpha)\geq 2\) by Dirichlet's theorem, whereas for irrational algebraic \(\alpha\) of degree \(d\), we have \(E(\alpha)\leq d\) by Liouville's theorem and \(E(\alpha)=2\) by Roth's theorem.NEWLINENEWLINEIn the sequel exactly the same definitions are applied for \textit{functions fields}, object of this article where the role of \(\mathbb Z, \mathbb Q, \mathbb R\) is played by \(A=\mathbb F[t], K=\mathbb F(t), K_\infty=\mathbb F((1/t))\) respectively, where \(\mathbb F\) is a finite field of characteristic \(p\) and the usual absolute value come from the degree in \(t\) of polynomials or rational functions. By rationals, reals and algebraic, we mean elements of \(K, K_\infty\) and algebraic over \(K\) respectively. Analogues of Dirichlet and Liouville theorems hold, But a \textit{naive} analog of Roth's theorem fails, as shown by \textit{K. Mahler} [Can. J. Math. 1, 397--400 (1949; Zbl 0033.35203)].NEWLINENEWLINENEWLINENEWLINEResults: In [\textit{D. Thakur}, Proc. Am. Math. Soc. 139, No. 1, 11--19 (2011; Zbl 1222.11089)], for \(p=2\) and any integer \(m>1\), the author constructed algebraic elements \(\alpha\) of degree at most \(2^{m^2}\) having continued fractions with folding pattern symmetries and a bounded sequence of partial quotients, so that \(E(\alpha)=2\) but with \textit{large (so corresponding to very…
reviewed by: Roland Quême
description: scientific article; zbMATH DE number 6184459