More on inhomogeneous diophantine approximation

Christopher G. Pinner

doi:10.5802/jtnb.337

Christopher G. Pinner

Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 539-557.

Abstract
Résumé

For an irrational real number $α$ and real number $γ$ we consider the inhomogeneous approximation constant $M (α, γ) : = \underset{| n | \to \infty}{lim inf} | n | | | n α - γ | |$ via the semi-regular negative continued fraction expansion of $α$ $α = \frac{1}{a_{1} - \frac{1}{a_{2} - \frac{1}{a_{3} - \dots}}}$ and an appropriate alpha-expansion of $γ$ . We give an upper bound on the case of worst inhomogeneous approximation, $ρ (α) : = sup_{γ \notin 𝐙 + α 𝐙} M (α, γ),$ which is sharp when the partial quotients ai are almost all even and at least four. When the negative expansion has period one we give a complete description of the spectrum of values $L (α) : = {M (α, γ) : γ \notin 𝐙 + α 𝐙},$ above the first limit point.

Pour un nombre irrationnel $α$ et un nombre réel $γ$ , on considère la constante d’approximation non-homogène $M (α, γ) : = \underset{| n | \to \infty}{lim inf} | n | | | n α - γ | |$ en rapport avec le développement en fraction continue négatif semi-régulier de $α$ $α = \frac{1}{a_{1} - \frac{1}{a_{2} - \frac{1}{a_{3} - \dots}}}$ et un $α$ -développement adéquat de $γ$ . Nous donnons une majoration de $ρ (α) : = sup_{γ \notin 𝐙 + α 𝐙} M (α, γ),$ dans le cas où $α$ est mal approximé, qui s’avère fine lorsque les quotients partiels $a_{i}$ sont presque tous pairs et supérieurs ou égaux à $4$ . Lorsque le développement de $α$ est de période $1$ , on décrit entièrement le spectre des valeurs prises par $𝐋 (α) : = {M (α, γ) : γ \notin 𝐙 + α 𝐙},$ au-dessus du premier point d’accumulation.

MR: 1879672 | Zbl: 1014.11043

DOI: 10.5802/jtnb.337

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     author = {Christopher G. Pinner},
     title = {More on inhomogeneous diophantine approximation},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {539--557},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     doi = {10.5802/jtnb.337},
     zbl = {1014.11043},
     mrnumber = {1879672},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.337/}
}

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Christopher G. Pinner. More on inhomogeneous diophantine approximation. Journal de Théorie des Nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 539-557. doi : 10.5802/jtnb.337. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.337/

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Cited by

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