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1. CJM 2007 (vol 59 pp. 211)
On Two Exponents of Approximation Related to a Real Number and Its Square 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 $|x_0| \le X$,
$|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le
X^{-\lambda}$ 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
$[\bQ(\xi)\wcol\bQ]>2$.
Categories:11J13, 11J82 |