Restricted rational approximation and Apéry-type constructions
Some definitions: Let us denote by \(\mathcal E\) the set of all functions \(\phi:\mathbb N^*\rightarrow\mathbb N^*\) with \(\mathbb N^*=\{1,2,3,\dots\}\) such that: 1) For any \(q\geq 1\), \(\phi(q+1)\) is a multiple of \(\phi(q)\). 2) The limit \(\gamma_\phi:=\lim_{q\rightarrow\infty}\frac{\log\;\phi(q)}{\log\;q}\) exists and satisfies \(0\leq \gamma_\phi<1\). For \(\phi\in\mathcal E\) and \(\xi\in\mathbb R\backslash \mathbb Q\), the \textit{\(\phi\)-exponent of irrationality} of \(\xi\) is the supremum denoted by \(\mu_\phi(\xi)\), of the set of real numbers \(\mu\) for which there are infinitely many \(q\geq 1\) such that \(q\) is a multiple of \(\phi(q)\) and \(|\xi-\frac{p}{q}|\leq \frac{1}{q^\mu}\) for some \(p\in\mathbb Z\). Observe that this definition generalizes the usual exponent of irrationality \(\mu(\xi)\). In this article the author obtains some general results on this exponent(a lower bound, the Hausdorff dimension of the set where it is large, connections to some linear linear form in \(1\) and \(\xi\), \(\dots\)). \textit{The context:} \textit{R. Apéry} has proved [Astérisque 61, 11--13 (1979; Zbl 0401.10049)] that, for \(\xi=\zeta(3), \alpha=e^3(1+\sqrt{2})^{-4}<1\), and \(\beta=e^3(1+\sqrt{2})^4>1\), the following holds: S.1: There exist two integer sequences \((u_n)_{n\geq 1}\) and \((v_n)_{n\geq 1}\) such that \(u_n\geq 0, |u_n\xi-v_n|^{1/n}\rightarrow \alpha\) and \(u_n^{1/n}\rightarrow \beta\). S.2: For \(d_n:=lcm(1,2,\dots,n)\), \(\delta_n:=d_n^3\) divides \(u_n\) for any \(n\geq 1\). S.3: S.1 implies \(\mu(\xi)\leq 1- \frac{\log\;\beta}{\log\;\alpha}\), where \(\mu(\xi)\) is the coefficient of irrationality of \(\xi\), so that \(\mu(\zeta(3))\leq 13.4178202\cdots\). S.4: \textit{S. Fischer, T. Rivoal} [Proc. Am. Math. Soc. 138, No. 3, 799--808 (2010; Zbl 1222.11094)] have proved that, for \(\xi\in \mathbb R\backslash\mathbb Q\), if \(\mu(\xi)\leq 1-\frac{\log\;\beta}{\log\;\alpha}\) with \(0<\alpha<1<\beta\), then \(S.1\) holds. \textit{The results:} The following is a generalization of S.3 and S.4: Theorem: Let \(\xi\in\mathbb R\backslash \mathbb Q,\;0<\alpha<1,\;\beta>1\), and \((\delta_n)_{n\geq 1}\) be a sequence of positive integers such that \(\delta_n\) divides \(\delta_{n+1}\) for any \(n\geq 1\), and \(\delta_n^{1/n}\rightarrow \delta\) as \(n\rightarrow \infty\). Define a function \(\phi\in\mathcal E\) by \(\phi(q)=\delta_n\) with \(n=\left[\frac{\log\;q}{\log(\delta/\alpha)}\right]\). Then we have the following implications: If S.1 and S.2 hold then \(\mu_\phi(\xi)\leq \frac{\log\;\beta-\log\;\alpha}{\log\;\delta -\log\;\alpha}\). If \(\mu_\phi(\xi)<\frac{\log\;\beta-\log\;\alpha}{\log\;\delta-\log\;\alpha}\) then S.1 and S.2 hold. This allows the author to refine a non effective result of \textit{A. K. Dubitskas} [Mosc. Univ. Math. Bull. 45, No. 2, 45--47 (1990); translation from Vestn. Mosk. Univ., Ser. I 1990, No. 2, 69--71 (1990; Zbl 0719.11039)]: Theorem: There exists \(c>0\) such that for any \(q\geq 1\) and any \(p\in\mathbb Z\) we have \[ \left|\zeta(3)-\frac{p}{q}\right|\geq \frac{c(\log\;q)^3}{q^2} \] provided that \(d_n^3:= \text{lcm}(1,2,\dots,n)^3\) divides \(q\), with \(n=\left[\frac{\log q}{\log((1\sqrt{2})^{4})}\right]\).
scientific article; zbMATH DE number 5691187
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