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

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 $\theta ^4=\theta ^{2}+1$.