Diophantine approximation and special Liouville numbers (Q2844468)

This paper introduces some methods to determine the simultaneous approximation constants of a class of \textit{well approximable numbers} \(\zeta_1,\zeta_2,\dots,\zeta_k\). The approach relies on results on the connection between the set of all \(s\)-adic expansions (\(s\geq 2)\) of \(\zeta_1, \zeta_2, \dots,\zeta_k\) and their associated approximation constants. As an application, explicit construction of real numbers \(\zeta_1,\zeta_2,\dots,\zeta_k\) with prescribed approximation properties are deduced and illustrated by \textit{Matlab} plots. (Author abstract).NEWLINENEWLINEMore precisely, the paper deals with the one parameter simultaneous approximation problem NEWLINE\[NEWLINE |x| \leq Q^{1+\theta},\;|\zeta_1 x-y_1| \leq Q^{-\frac{1}{k}+\theta}, \dots, |\zeta_kx-y_k|\leq Q^{-\frac{1}{k}+\theta},NEWLINE\]NEWLINE where \(\zeta_1,\dots,\zeta_k\) are real numbers which are assumed to be linearly independent together with \(1\) and \(x,y_1,\dots,y_k\) are integers to be determined in dependence of the parameter \(Q>1\) in order to minimize \(\theta\).NEWLINENEWLINEFor fixed \(\zeta_1,\dots,\zeta_k\) and for every \(X>0\) define the functions \(\omega_j(X)\) as the supremum over all real numbers \(\nu\) such that the system \(|x|\leq X, \;|\zeta_i x-y_i|\leq X^{-\nu},\;1\leq i\leq k\;\) has \(j\) linearly independent solutions \((x,y_1,\dots,y_k)\in\mathbb Z^{k+1}\). Following [\textit{V. Jarník}, Czech. Math. J. 4(79), 330--353 (1954; Zbl 0057.28303)], the \textit{classical approximation constant} \(\omega_j, \hat{\omega}_j\) are defined by \(\omega_j=\lim \sup_{X\rightarrow\infty} \omega_j(X)\) and \(\hat{\omega}_j=\lim \inf_{X\rightarrow\infty} \omega_j(X).\) The author proves:NEWLINENEWLINE{Theorem. } Let \(k\) be a positive integer and for \(1\leq j\leq k\) let \(\zeta_j=\sum_{n\geq 1} \frac{1}{q_n,j}\) where \(q_{1,1}<q_{1,2}<\dots <q_{1,k}<q_{2,1}\dots <q_{2,k}<q_{3,1},\dots\) are natural integers such that \(q_{n,j}| q_{n,j+1}\) for \(1\leq j\leq k-1\) and \(q_{n,k}|q_{n+1,1}\) for all \(n\geq 1\) and such that NEWLINE\[NEWLINE\begin{split} &\lim_{n\rightarrow\infty }(\log (q_{n+1,1})-\log (q_{n,k}))/\log (q_{n+1,k})=\eta_1,\\ &\lim_{n\rightarrow\infty }(\log (q_{n+1,i})-\log (q_{n+1,i-1}))/\log (q_{n+1,k})=\eta_i,\;2\leq i\leq k,\\ &\lim_{n\rightarrow\infty}(\log (q_{n+1,1}))/\log (q_{n,k})=\eta_{k+1}=\infty,\\ \end{split}NEWLINE\]NEWLINE where \(\eta=(\eta_1,\dots,\eta_{k+1})\in\mathbb R^k\times \bar \mathbb R\) satisfy \(\eta_1+\dots+\eta_k=1\), \(\;\eta_{k+1}>\eta_k\geq \eta_{k-1}\geq\dots\geq \eta_1>0,\;\eta_{k+1}=\infty\).NEWLINENEWLINESuppose that \(\omega_1=\infty\). Then the classical approximation constants \(\omega_j\) relative to the vector \((\zeta_1,\dots,\zeta_k)\) are given by \(\omega_1=\infty, \;\omega_2=\max \Big\{\frac{\eta_k}{\eta_k+\dots+\eta_1}, \frac{\eta_{k-1}}{\eta_{k-1}+\dots+\eta_1},\dots ,\frac{\eta_1}{\eta_1}\Big\},\dots \) \(\omega_3=\max \Big\{\frac{\eta_{k-1}}{\eta_k+\dots+\eta_1}, \frac{\eta_{k-2}}{\eta_{k-1}+\dots+\eta_1},\dots \frac{\eta_1}{\eta_2+\eta_1}\Big\},\;\) \(\omega_{k+1}=\frac{\eta_1}{\eta_k+\dots+\eta_1}\) and the constant \(\hat{\omega}_j\) are given by \(\hat{\omega}_1=\min \Big\{\frac{\eta_k}{\eta_k+\dots+\eta_1}, \frac{\eta_{k-1}}{\eta_{k-1}+\dots+\eta_1},\dots ,\frac{\eta_1}{\eta_1}\Big\},\;\) \(\hat{\omega}_j=0, \;2\leq j\leq k+1\).NEWLINENEWLINE The author gives also a theorem of same nature if \(\hat{\omega}_1=\infty\) and some example which satisfies a conjecture of [\textit{W. M. Schmidt} and \textit{L. Summerer}, Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)].