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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators
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by J. C. Lagarias PDF
Trans. Amer. Math. Soc. 272 (1982), 545-554 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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