# More on inhomogeneous diophantine approximation

• Volume: 13, Issue: 2, page 539-557
• ISSN: 1246-7405

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## Abstract

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For an irrational real number$\alpha$ and real number $\gamma$ we consider the inhomogeneous approximation constant$M\left(\alpha ,\gamma \right):=\underset{|n|\to \infty }{lim inf}|n|||n\alpha -\gamma ||$via the semi-regular negative continued fraction expansion of $\alpha$$\alpha =\frac{1}{{a}_{1}-\frac{1}{{a}_{2}-\frac{1}{{a}_{3}-\cdots }}}$and an appropriate alpha-expansion of $\gamma$. We give an upper bound on the case of worst inhomogeneous approximation,$\rho \left(\alpha \right):=\underset{\gamma \notin 𝐙+\alpha 𝐙}{sup}M\left(\alpha ,\gamma \right),$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$L\left(\alpha \right):=\left\{M\left(\alpha ,\gamma \right):\gamma \notin 𝐙+\alpha 𝐙\right\},$above the first limit point.

## How to cite

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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 -

## References

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1. [1] E.S. Barnes, H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms. Part I, Acta Math.87 (1952), 259-323; Part II, Acta Math.88 (1952), 279-316; Part III, Acta Math.92 (1954), 199-234; Part IV (without second author) Acta Math.92 (1954), 235-264. Zbl0056.27301
2. [2] T.W. Cusick, A.M. Rockett, P. Szúsz, On inhomogeneous Diophantine approximation. J. Number Theory48 (1994), 259-283. Zbl0820.11042MR1293862
3. [3] H. Davenport, Non-homogeneous binary quadratic forms. Nederl. Akad. Wetensch. Proc.50 (1947), 741-749, 909-917 = Indagationes Math.9 (1947), 351-359, 420-428. Zbl0060.11906MR23856
4. [4] T. Komatsu, On inhomogeneous diophantine approximation and the Nishioka - Shiokawa- Tamura algorithm. Acta Arith.86 (1998), 305-324. Zbl0930.11049MR1659089
5. [5] W. Moran, C. Pinner, A. Pollington, On inhomogeneous Diophantine approximation, preprint.
6. [6] P. Varnavides, Non-homogeneous quadratic forms, I, II. Nederl. Akad. Wetensch. Proc.51, (1948) 396-404, 470-481. = Indagationes Math.10 (1948), 142-150, 164-175. Zbl0030.01901MR25517

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