More on inhomogeneous diophantine approximation
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, p. 539-557
Pour un nombre irrationnel α et un nombre réel γ, on considère la constante d’approximation non-homogène M(α,γ):=lim inf |n| |n|||nα-γ|| en rapport avec le développement en fraction continue négatif semi-régulier de α α=1 a 1 -1 a 2 -1 a 3 - et un α-développement adéquat de γ. Nous donnons une majoration de ρ(α):=sup γ𝐙+α𝐙 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):γ𝐙+α𝐙}, au-dessus du premier point d’accumulation.
For an irrational real numberα and real number γ we consider the inhomogeneous approximation constant M(α,γ):=lim inf |n| |n|||nα-γ|| via the semi-regular negative continued fraction expansion of α α=1 a 1 -1 a 2 -1 a 3 - and an appropriate alpha-expansion of γ. We give an upper bound on the case of worst inhomogeneous approximation, ρ(α):=sup γ𝐙+α𝐙 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):γ𝐙+α𝐙}, above the first limit point.
@article{JTNB_2001__13_2_539_0,
     author = {Pinner, Christopher G.},
     title = {More on inhomogeneous diophantine approximation},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     pages = {539-557},
     zbl = {1014.11043},
     mrnumber = {1879672},
     language = {en},
     url = {http://http://www.numdam.org/item/JTNB_2001__13_2_539_0}
}
Pinner, Christopher G. More on inhomogeneous diophantine approximation. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 539-557. http://www.numdam.org/item/JTNB_2001__13_2_539_0/

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