More on inhomogeneous diophantine approximation

Pinner, Christopher G.. "More on inhomogeneous diophantine approximation." Journal de théorie des nombres de Bordeaux 13.2 (2001): 539-557. <http://eudml.org/doc/248712>.

@article{Pinner2001,
abstract = {For an irrational real number$\alpha $ and real number $\gamma $ we consider the inhomogeneous approximation constant\begin\{equation*\} M(\alpha , \gamma ) := \liminf \_\{|n| \rightarrow \infty \} |n|||n \alpha - \gamma || \end\{equation*\}via the semi-regular negative continued fraction expansion of $\alpha $\begin\{equation*\} \alpha = \cfrac\{1\}\{a\_1- \cfrac\{1\}\{a\_2- \cfrac\{1\}\{a\_3- \cdots \}\}\} \end\{equation*\}and an appropriate alpha-expansion of $\gamma $. We give an upper bound on the case of worst inhomogeneous approximation,\begin\{equation*\} \rho (\alpha ) := \sup \_\{\gamma \notin \mathbf \{Z\}+ \alpha \mathbf \{Z\}\} M(\alpha , \gamma ), \end\{equation*\}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\begin\{equation*\} L(\alpha ) := \lbrace M(\alpha , \gamma ) : \gamma \notin \mathbf \{Z\} + \alpha \mathbf \{Z\} \rbrace , \end\{equation*\}above the first limit point.},
author = {Pinner, Christopher G.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {inhomogeneous approximation constants; semi-regular continued fraction; one-sided approximation spectrum; backward continued fraction expansion; -expansion},
language = {eng},
number = {2},
pages = {539-557},
publisher = {Université Bordeaux I},
title = {More on inhomogeneous diophantine approximation},
url = {http://eudml.org/doc/248712},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Pinner, Christopher G.
TI - More on inhomogeneous diophantine approximation
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 539
EP - 557
AB - For an irrational real number$\alpha $ and real number $\gamma $ we consider the inhomogeneous approximation constant\begin{equation*} M(\alpha , \gamma ) := \liminf _{|n| \rightarrow \infty } |n|||n \alpha - \gamma || \end{equation*}via the semi-regular negative continued fraction expansion of $\alpha $\begin{equation*} \alpha = \cfrac{1}{a_1- \cfrac{1}{a_2- \cfrac{1}{a_3- \cdots }}} \end{equation*}and an appropriate alpha-expansion of $\gamma $. We give an upper bound on the case of worst inhomogeneous approximation,\begin{equation*} \rho (\alpha ) := \sup _{\gamma \notin \mathbf {Z}+ \alpha \mathbf {Z}} M(\alpha , \gamma ), \end{equation*}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\begin{equation*} L(\alpha ) := \lbrace M(\alpha , \gamma ) : \gamma \notin \mathbf {Z} + \alpha \mathbf {Z} \rbrace , \end{equation*}above the first limit point.
LA - eng
KW - inhomogeneous approximation constants; semi-regular continued fraction; one-sided approximation spectrum; backward continued fraction expansion; -expansion
UR - http://eudml.org/doc/248712
ER -