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ON THE SPECTRUM OF DIOPHANTINE APPROXIMATION CONSTANTS

Part of:Diophantine approximation, transcendental number theoryGeometry of numbers

Published online by Cambridge University Press:  22 June 2015

Johannes Schleischitz
Johannes Schleischitz*
Affiliation:
Institute of Mathematics, University of Natural Resources and Life Sciences, Vienna, Austria email johannes.schleischitz@boku.ac.at

Abstract

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The approximation constant ${\it\lambda}_{k}({\it\zeta})$ is defined as the supremum of ${\it\eta}\in \mathbb{R}$ such that the estimate $\max _{1\leqslant j\leqslant k}\Vert {\it\zeta}^{j}x\Vert \leqslant x^{-{\it\eta}}$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\it\lambda}_{1}({\it\zeta}),{\it\lambda}_{2}({\it\zeta}),\ldots )$, which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\it\lambda}_{k}({\it\zeta})$ as well. In particular, for given $k\geqslant 1$ and ${\it\lambda}\geqslant 1$, we construct ${\it\zeta}$ in the Cantor set with ${\it\lambda}_{k}({\it\zeta})={\it\lambda}$. Moreover, we establish an estimate for the uniform approximation constants $\widehat{{\it\lambda}}_{k}({\it\zeta})$, which enables us to determine classical approximation constants for Liouville numbers.

MSC classification

Primary:11H06: Lattices and convex bodies
Secondary:11J13: Simultaneous homogeneous approximation, linear forms11J25: Diophantine inequalities11J82: Measures of irrationality and of transcendence11J83: Metric theory

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016, pp. 79 - 100
Copyright
Copyright © University College London 2015 

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