More on inhomogeneous diophantine approximation
Christopher G. Pinner
Journal de Théorie des Nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 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 γ∉𝐙+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):γ∉𝐙+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 γ∉𝐙+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):γ∉𝐙+α𝐙}, above the first limit point.

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     author = {Pinner, Christopher G.},
     title = {More on inhomogeneous diophantine approximation},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {539--557},
     publisher = {Universit\'e Bordeaux I},
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Christopher G. Pinner. More on inhomogeneous diophantine approximation. Journal de Théorie des Nombres de Bordeaux, Tome 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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