An approximation property of quadratic irrationals

Takao Komatsu

Bulletin de la Société Mathématique de France (2002)

  • Volume: 130, Issue: 1, page 35-48
  • ISSN: 0037-9484

Abstract

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Let α > 1 be irrational. Several authors studied the numbers ℓ m ( α ) = inf { | y | : y ∈ Λ m , y ≠ 0 } , where m is a positive integer and Λ m denotes the set of all real numbers of the form y = ϵ 0 α n + ϵ 1 α n - 1 + ⋯ + ϵ n - 1 α + ϵ n with restricted integer coefficients | ϵ i | ≤ m . The value of ℓ 1 ( α ) was determined for many particular Pisot numbers and ℓ m ( α ) for the golden number. In this paper the value of  ℓ m ( α ) is determined for irrational numbers  α , satisfying α 2 = a α ± 1 with a positive integer a .

How to cite

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Komatsu, Takao. "An approximation property of quadratic irrationals." Bulletin de la Société Mathématique de France 130.1 (2002): 35-48. <http://eudml.org/doc/272476>.

@article{Komatsu2002,
abstract = {Let $\alpha &gt;1$ be irrational. Several authors studied the numbers\[ \{\ell ^m(\alpha )=\inf \lbrace \, |y|:y\in \Lambda \_m,\, y\ne 0\rbrace \}, \]where $m$ is a positive integer and $\Lambda _m$ denotes the set of all real numbers of the form $y=\epsilon _0\alpha ^n+\epsilon _1\alpha ^\{n-1\}+\cdots +\epsilon _\{n-1\}\alpha +\epsilon _n$ with restricted integer coefficients $|\epsilon _i|\le m$. The value of $\ell ^1(\alpha )$ was determined for many particular Pisot numbers and $\ell ^m(\alpha )$ for the golden number. In this paper the value of $\ell ^m(\alpha )$ is determined for irrational numbers $\alpha $, satisfying $\alpha ^2=a\alpha \pm 1$ with a positive integer $a$.},
author = {Komatsu, Takao},
journal = {Bulletin de la Société Mathématique de France},
keywords = {approximation property; quadratic irrationals; continued fractions},
language = {eng},
number = {1},
pages = {35-48},
publisher = {Société mathématique de France},
title = {An approximation property of quadratic irrationals},
url = {http://eudml.org/doc/272476},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Komatsu, Takao
TI - An approximation property of quadratic irrationals
JO - Bulletin de la Société Mathématique de France
PY - 2002
PB - Société mathématique de France
VL - 130
IS - 1
SP - 35
EP - 48
AB - Let $\alpha &gt;1$ be irrational. Several authors studied the numbers\[ {\ell ^m(\alpha )=\inf \lbrace \, |y|:y\in \Lambda _m,\, y\ne 0\rbrace }, \]where $m$ is a positive integer and $\Lambda _m$ denotes the set of all real numbers of the form $y=\epsilon _0\alpha ^n+\epsilon _1\alpha ^{n-1}+\cdots +\epsilon _{n-1}\alpha +\epsilon _n$ with restricted integer coefficients $|\epsilon _i|\le m$. The value of $\ell ^1(\alpha )$ was determined for many particular Pisot numbers and $\ell ^m(\alpha )$ for the golden number. In this paper the value of $\ell ^m(\alpha )$ is determined for irrational numbers $\alpha $, satisfying $\alpha ^2=a\alpha \pm 1$ with a positive integer $a$.
LA - eng
KW - approximation property; quadratic irrationals; continued fractions
UR - http://eudml.org/doc/272476
ER -

References

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  1. [1] Y. Bugeaud – « On a property of Pisot numbers and related questions », Acta Math. Hungar.73 (1996), p. 33–39. Zbl0923.11148MR1415918
  2. [2] M. Djawadi & G. Hofmeister – « Linear diophantine problems », Arch. Math. (Basel) 66 (1996), p. 19–29. Zbl0854.11016MR1363774
  3. [3] P. Erdős, I. Joó & M. Joó – « On a problem of Tamás Varga », Bull. Soc. Math. France120 (1992), p. 507–521. Zbl0787.11002MR1194274
  4. [4] P. Erdős, I. Joó & V. Komornik – « Characterization of the unique expressions 1 = ∑ q - n i and related problems », Bull. Soc. Math. France118 (1990), p. 377–390. Zbl0721.11005MR1078082
  5. [5] —, « On the sequence of numbers of the form ϵ 0 + ϵ 1 q + ⋯ + ϵ n q n , ϵ i ∈ { 0 , 1 } », Acta Arith.83 (1998), p. 201–210. Zbl0896.11006MR1611185
  6. [6] P. Erdős & V. Komornik – « On developments in noninteger bases », Acta Math. Hungar.79 (1998), p. 57–83. Zbl0906.11008MR1611948
  7. [7] V. Komornik & P. Loreti – « Unique developments in non-integer bases », Amer. Math. Monthly105 (1998), p. 636–639. Zbl0918.11006MR1633077
  8. [8] V. Komornik, P. Loreti & M. Pedicini – « An approximation property of Pisot numbers », J. Number Theory80 (2000), p. 218–237. Zbl0962.11034MR1740512
  9. [9] T. van Ravenstein – « The three gap theorem (Steinhaus conjecture) », J. Austral. Math. Soc. Ser. A45 (1988), p. 360–370. Zbl0663.10039MR957201
  10. [10] W. Schmidt – Diophantine approximation, Lecture Notes in Math., vol. 785, Springer-Verlag, 1980. Zbl0421.10019MR568710

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