Simultaneous rational approximation via Rickert's integrals
In the literature simultaneous rational approximation to algebraic numbers has been studied extensively. The authors focus on the method using hypergeometric polynomials as Padé approximants to functions of the form \[ (1+ax)^{\nu},\;a\in\mathbb{Z}\setminus\{0\},\;\nu\in\mathbb Q^{+}\setminus\mathbb N. \] An important tool is a contour integral representation of these approximants due to \textit{J. H. Rickert} [Math. Proc. Camb. Philos. Soc. 113, No. 3, 461--472 (1993; Zbl 0786.11040)]. The authors apply this tool in the following setting. Let \(a\), \(N\) be positive integers, \(0<\nu<1\) rational and write \(\nu=s/n\) with \(\text{gcd}(s,n)=1\). Put \[ e_0=\prod_{p|n}\,p^{\max\{\text{ord}_p(n/2a)+1/(p-1),0\}}, \] \[ (e,e',e'')\;\begin{cases}(2,4/3,168) &\ \text{if}\ n=2\\ (4,3,679) &\ \text{if}\ n=4\\ (e_0,3e_0/2,1) &\ \text{otherwise}\end{cases} \] Theorem 1.1. Let \(\theta_1=(1-{a\over N})^{\nu},\;\theta_2=(1+{a\over N})^{\nu}\) with \(N\geq 6.794 (e')^2a^3\) and assume that at least one of \(\theta_1\), \(\theta_2\) is irrational. Then \[ \max\{\left|\theta_1-{p_1\over q}\right|,\left|\theta_2-{p_2\over q}\right|\}>{c\over 9^{ \lambda}}\eqno{(\ast)} \] for all integers \(p_1\), \(p_2\), \(q\) with \(q>0\), where \[ \lambda=1+{\log{(6\sqrt{3}e'N+12ae')}\over \log{(27(N^2-a^2)/(16a^3e'))}} \] and \[ c={0.005\over ae'e''N}\left(1-{a\over N}\right)^{\nu}\left(1+{2a\over N\sqrt{3}}\right)^{-1-\nu} \left({qNae'' \nu (1-\nu^2)\over 2(N-a)}\right)^{ -\lambda+1}. \] Theorem 1.3. Let \(\theta_1=(1+{a\over N})^{\nu},\;\theta_2=(1+{2a\over N})^{\nu}\) with \(N\geq 8.637 (e')^2a^3\) and assume that at least one of \(\theta_1\), \(\theta_2\) is irrational. Then \((\ast)\) holds with \[ \lambda=1+{\log{(6\sqrt{3}e'N+6ae'(2+\sqrt{3})}\over \log{(27(N-a)(N-2a)/(16a^3e'))}} \] and \[ c={0.005\over ae'e''N}\left(1+{(2+\sqrt{3})a\over N\sqrt{3}}\right)^{-1-\nu} \left({9Nae'' \nu (1-\nu^2)\over 2(N-2a)}\right)^{ -\lambda+1}. \] The third main result (Theorem 1.6) runs along the same type of approach, but needs to many quantities to be introduced, to cite it explicitly here.
scientific article; zbMATH DE number 6374449