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SIMULTANEOUS APPROXIMATION TO TWO REALS: BOUNDS FOR THE SECOND SUCCESSIVE MINIMUM

Part of: Diophantine approximation, transcendental number theory Geometry of numbers

Published online by Cambridge University Press:  29 November 2017

Wolfgang M. Schmidt  and
Leonhard Summerer
Wolfgang M. Schmidt
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, U.S.A. email [email protected]
Leonhard Summerer
Affiliation:
Fakultät für Mathematik der Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria email [email protected]
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Abstract

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$, $(i=1,2,3)$, attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$, give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “$3$-systems”.

MSC classification

Primary: 11H06: Lattices and convex bodies 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 1136 - 1151
Copyright
Copyright © University College London 2017 

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References

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