For each real number
$\xi$
, let
$\widehat{{{\lambda }_{2}}}\left( \xi \right)$
denote the supremum of all real numbers
$\text{ }\!\!\lambda\!\!\text{ }$
such that, for each sufficiently large
$X$
, the inequalities
$\left| {{x}_{0}} \right|\,\le \,X,\,\left| {{x}_{0}}\xi \,-\,{{x}_{1}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$
and
$\left| {{x}_{0}}{{\xi }^{2}}\,-\,{{x}_{2}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$
admit a solution in integers
${{x}_{0}},\,{{x}_{1}}$
and
${{x}_{2}}$
not all zero, and let
$\widehat{{{\omega }_{2}}}\left( \xi \right)$
denote the supremum of all real numbers
$\omega $
such that, for each sufficiently large
$X$
, the dual inequalities
$\left| {{x}_{0}}\,+\,{{x}_{1}}\xi \,+\,{{x}_{2}}{{\xi }^{2}} \right|\,\le \,{{X}^{-\omega }}$
,
$\left| {{x}_{1}} \right|\,\le \,X$
and
$\left| {{x}_{2}} \right|\,\le \,X$
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
$\widehat{{{\lambda }_{2}}}\left( \xi \right)$
where
$\xi$
ranges through all real numbers with
$[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$
form a dense subset of the interval
$\left[ 1/2,\,\left( \sqrt{5}\,-\,1 \right)/2 \right]$
while, for the same values of
$\xi$
, the dual exponents
$\widehat{{{\omega }_{2}}}\left( \xi \right)$
form a dense subset of
$\left[ 2,\,\left( \sqrt{5}\,+\,3 \right)/2 \right]$
. Part of the proof rests on a result of V. Jarník showing that
$\widehat{{{\lambda }_{2}}}\left( \xi \right)=1-{{\hat{\omega }}_{2}}{{\left( \xi \right)}^{-1}}$
for any real number
$\xi$
with
$[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$
.