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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
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\Vert \,\cdot\, \right\Vert $ on $ R^n$. Suppose $ \alpha$ is not rational and order the best approximations to $ \alpha$ with respect to $ \left\Vert\, \cdot\, \right\Vert $ 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

$\displaystyle g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right) = ... ...liminf\limits_{k \to \infty }} {({q_k})^{1/k}} \geq 1 + \frac{1}{{{2^{n + 1}}}}$

. Let $ g\left( {\left\Vert\, \cdot\, \right\Vert} \right)$ denote the infimum of $ g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right)$ over all $ \alpha$ in $ R^n$ with an irrational coordinate. For the sup norm $ \left\Vert\, \cdot \,\right\Vert _s$ on $ R^2$ it is shown that $ g\left( {\left\Vert \, \cdot \, \right\Vert}_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