Higher Diophantine approximation exponents and continued fraction symmetries for function fields. II
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 good quadratic approximations)} \(E_2(\alpha)\geq 2^m>3\). In this paper, with a different construction, based on [\textit{D. Thakur}, J. Number Theory 79, No. 2, 284--291 (1999; Zbl 0966.11029), and the above cited paper] generalizing this result, he proves:NEWLINENEWLINELet \(p\) be a prime, \(q\) be a power of \(p\) and \(\varepsilon>0\) be given. Then it is possible to construct infinitely many algebraic \(\alpha\), with explicit equations and continued fractions, such that NEWLINE\[NEWLINEq\leq \deg(\alpha)\leq q+1, \;E(\alpha)<2+\varepsilon,\;E_2(\alpha)>q-\varepsilon,NEWLINE\]NEWLINE with an explicit sequence of quadratic approximations realizing the last bound.NEWLINENEWLINELet \(p\) be a prime, \(q\) be a power of \(p\) and \(m,n>1\) be given. Then it is possible to contruct infinitely many algebraic \(\alpha_{m,n}\), with explicit equations and continued fractions, such that NEWLINE\[NEWLINE\deg(\alpha_{m,n})\leq q^m+1,\;\lim_{n\rightarrow\infty} E(\alpha_{m,n})=2,\;\lim_{n\rightarrow\infty} E_{q+1}(\alpha_{m,n})\geq q^{m-1}+\frac{q-1}{(q+1)q},NEWLINE\]NEWLINE with an explicit sequence of degree \(q+1\)-approximations realizing the last bound.
scientific article; zbMATH DE number 6184459