Diophantine approximation on Veech surfaces

Pascal Hubert; Thomas A. Schmidt

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 4, page 551-568
  • ISSN: 0037-9484

Abstract

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We show that Y. Cheung’s general Z -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.

How to cite

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Hubert, Pascal, and Schmidt, Thomas A.. "Diophantine approximation on Veech surfaces." Bulletin de la Société Mathématique de France 140.4 (2012): 551-568. <http://eudml.org/doc/272605>.

@article{Hubert2012,
abstract = {We show that Y. Cheung’s general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.},
author = {Hubert, Pascal, Schmidt, Thomas A.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {translation surfaces; transcendence; diophantine approximation},
language = {eng},
number = {4},
pages = {551-568},
publisher = {Société mathématique de France},
title = {Diophantine approximation on Veech surfaces},
url = {http://eudml.org/doc/272605},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Hubert, Pascal
AU - Schmidt, Thomas A.
TI - Diophantine approximation on Veech surfaces
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 4
SP - 551
EP - 568
AB - We show that Y. Cheung’s general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments.
LA - eng
KW - translation surfaces; transcendence; diophantine approximation
UR - http://eudml.org/doc/272605
ER -

References

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