On simultaneous rational approximation to a real number and its integral powers

Yann Bugeaud[1]

  • [1] Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 6, page 2165-2182
  • ISSN: 0373-0956

Abstract

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For a positive integer n and a real number ξ , let λ n ( ξ ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that | | q ξ | | , | | q ξ 2 | | , ... , | | q ξ n | | are all less than q - λ . Here, | | · | | denotes the distance to the nearest integer. We study the set of values taken by the function λ n and, more generally, we are concerned with the joint spectrum of ( λ 1 , ... , λ n , ... ) . We further address several open problems.

How to cite

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Bugeaud, Yann. "On simultaneous rational approximation to a real number and its integral powers." Annales de l’institut Fourier 60.6 (2010): 2165-2182. <http://eudml.org/doc/116329>.

@article{Bugeaud2010,
abstract = {For a positive integer $n$ and a real number $\xi $, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda $ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ are all less than $q^\{-\lambda \}$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We study the set of values taken by the function $\lambda _n$ and, more generally, we are concerned with the joint spectrum of $(\lambda _1, \ldots , \lambda _n , \ldots )$. We further address several open problems.},
affiliation = {Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)},
author = {Bugeaud, Yann},
journal = {Annales de l’institut Fourier},
keywords = {Simultaneous approximation; exponent of approximation; simultaneous approximation},
language = {eng},
number = {6},
pages = {2165-2182},
publisher = {Association des Annales de l’institut Fourier},
title = {On simultaneous rational approximation to a real number and its integral powers},
url = {http://eudml.org/doc/116329},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Bugeaud, Yann
TI - On simultaneous rational approximation to a real number and its integral powers
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 6
SP - 2165
EP - 2182
AB - For a positive integer $n$ and a real number $\xi $, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda $ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ are all less than $q^{-\lambda }$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We study the set of values taken by the function $\lambda _n$ and, more generally, we are concerned with the joint spectrum of $(\lambda _1, \ldots , \lambda _n , \ldots )$. We further address several open problems.
LA - eng
KW - Simultaneous approximation; exponent of approximation; simultaneous approximation
UR - http://eudml.org/doc/116329
ER -

References

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