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On Two Exponents of Approximation Related to a Real Number and Its Square


Published:20070201
Printed: Feb 2007
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Abstract
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $x_0 \le X$,
$x_0\xix_1 \le X^{\lambda}$ and $x_0\xi^2x_2 \le
X^{\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $x_0+x_1\xi+x_2\xi^2 \le
X^{\omega}$, $x_1 \le X$ and $x_2 \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1\omegahat_2(\xi)^{1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.