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

Author:

Lagarias J. C.

Abstract

This paper defines the notion of a best simultaneous Diophantine approximation to a vector α \alpha in R n R^n with respect to a norm \left \| \,\cdot \, \right \| on R n R^n . Suppose α \alpha is not rational and order the best approximations to α \alpha with respect to \left \|\, \cdot \, \right \| by increasing denominators 1 = q 1 > q 2 > 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 \[ g ( α , ) = lim inf k ( q k ) 1 / k 1 + 1 2 n + 1 g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right ) = \liminf \limits _{k \to \infty } {({q_k})^{1/k}} \geq 1 + \frac {1}{{{2^{n + 1}}}} \] . Let g ( ) g\left ( {\left \|\, \cdot \, \right \|} \right ) denote the infimum of g ( α , ) g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right ) over all α \alpha in R n R^n with an irrational coordinate. For the sup norm s \left \|\, \cdot \,\right \|_s on R 2 R^2 it is shown that g ( s ) θ = 1.270 + g\left ( {\left \| \, \cdot \, \right \|}_s \right )\ge \theta =1.270^{+} where θ 4 = θ 2 + 1 \theta ^4=\theta ^{2}+1 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference17 articles.

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4. Formulas for some Diophantine approximation constants. II;Cusick, T. W.;Acta Arith.,1974

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