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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators


Author: J. C. Lagarias
Journal: Trans. Amer. Math. Soc. 272 (1982), 545-554
MSC: Primary 10F10; Secondary 10F20
DOI: https://doi.org/10.1090/S0002-9947-1982-0662052-7
MathSciNet review: 662052
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Abstract: This paper defines the notion of a best simultaneous Diophantine approximation to a vector $\alpha$ in $R^n$ with respect to a norm $\left \| \cdot \right \|$ on $R^n$. Suppose $\alpha$ is not rational and order the best approximations to $\alpha$ with respect to $\left \| \cdot \right \|$ by increasing denominators $1=q_1 < q_2 < \cdots$. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that \[ g\left ( {\alpha , \left \| { \cdot } \right \|} \right ) = \liminf \limits _{k \to \infty } {({q_k})^{1/k}} \geq 1 + \frac {1}{{{2^{n + 1}}}}\]. Let $g\left ( {\left \| \cdot \right \|} \right )$ denote the infimum of $g\left ( {\alpha , \left \| { \cdot } \right \|} \right )$ over all $\alpha$ in $R^n$ with an irrational coordinate. For the sup norm $\left \| \cdot \right \|_s$ on $R^2$ it is shown that $g\left ( {\left \| \cdot \right \|}_s \right )\ge \theta =1.270^{+}$ where $\theta ^4=\theta ^{2}+1$.


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Keywords: Simultaneous Diophantine approximation
Article copyright: © Copyright 1982 American Mathematical Society