Diophantine approximation and special Liouville numbers

Johannes Schleischitz

Communications in Mathematics (2013)

  • Volume: 21, Issue: 1, page 39-76
  • ISSN: 1804-1388

Abstract

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This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers ζ 1 , ζ 2 , ... , ζ k . The approach relies on results on the connection between the set of all s -adic expansions ( s ≥ 2 ) of ζ 1 , ζ 2 , ... , ζ k and their associated approximation constants. As an application, explicit construction of real numbers ζ 1 , ζ 2 , ... , ζ k with prescribed approximation properties are deduced and illustrated by Matlab plots.

How to cite

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Schleischitz, Johannes. "Diophantine approximation and special Liouville numbers." Communications in Mathematics 21.1 (2013): 39-76. <http://eudml.org/doc/260600>.

@article{Schleischitz2013,
abstract = {This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta _\{1\},\zeta _\{2\},\ldots ,\zeta _\{k\}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of $\zeta _\{1\},\zeta _\{2\},\ldots ,\zeta _\{k\}$ and their associated approximation constants. As an application, explicit construction of real numbers $\zeta _\{1\},\zeta _\{2\},\ldots ,\zeta _\{k\}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.},
author = {Schleischitz, Johannes},
journal = {Communications in Mathematics},
keywords = {convex geometry; lattices; Liouville numbers; successive minima; convex geometry; lattices; Liouville numbers; successive minima},
language = {eng},
number = {1},
pages = {39-76},
publisher = {University of Ostrava},
title = {Diophantine approximation and special Liouville numbers},
url = {http://eudml.org/doc/260600},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Schleischitz, Johannes
TI - Diophantine approximation and special Liouville numbers
JO - Communications in Mathematics
PY - 2013
PB - University of Ostrava
VL - 21
IS - 1
SP - 39
EP - 76
AB - This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta _{1},\zeta _{2},\ldots ,\zeta _{k}$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of $\zeta _{1},\zeta _{2},\ldots ,\zeta _{k}$ and their associated approximation constants. As an application, explicit construction of real numbers $\zeta _{1},\zeta _{2},\ldots ,\zeta _{k}$ with prescribed approximation properties are deduced and illustrated by Matlab plots.
LA - eng
KW - convex geometry; lattices; Liouville numbers; successive minima; convex geometry; lattices; Liouville numbers; successive minima
UR - http://eudml.org/doc/260600
ER -

References

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  1. Gruber, P.M., Lekkerkerker, C.G., Geometry of numbers, 1987, North-Holland Verlag, (1987) Zbl0611.10017MR0893813
  2. Jarník, V., Contribution to the theory of linear homogeneous diophantine approximation, Czechoslovak Math. J., 4, 79, 1954, 330-353, (1954) MR0072183
  3. Moshchevitin, N.G., Proof of W.M. Schmidt's conjecture concerning successive minima of a lattice, J. London Math. Soc., 2012, doi: 10.1112/jlms/jdr076, 12 Mar 2012. (2012) MR2959298
  4. Roy, D., Diophantine approximation in small degree, 2004, Number theory 269--285, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI. (2004) Zbl1077.11051MR2076601
  5. Schmidt, W.M., Summerer, L., 10.4064/aa140-1-5, Acta Arithm., 140, 1, 2009, (2009) Zbl1236.11060MR2557854DOI10.4064/aa140-1-5
  6. Schmidt, W.M., Summerer, L., Diophantine approximation and parametric geometry of numbers, to appear in Monatshefte für Mathematik. Zbl1264.11056MR3016519
  7. Waldschmidt, M., Report on some recent advances in Diophantine approximation, 2009, (2009) 

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