For any pair
$i,j\ge 0$
with
$i+j=1$
let
${\mathbf Bad}(i,j)$
denote the set of pairs
$(\alpha,\beta)\in {\bb R}^2$
for which
$\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$
for all
$q\in {\bb N}$
. Here
$c=c(\alpha,\beta)$
is a positive constant. If
$i=0$
the set
${\mathbf Bad}(0, 1)$
is identified with
${\bb R}\times {\mathbf Bad}$
where
${\mathbf Bad}$
is the set of badly approximable numbers. That is,
${\mathbf Bad}(0, 1)$
consists of pairs
$(\alpha, \beta)$
with
$\alpha\in {\bb R}$
and
$\beta\in {\mathbf Bad}$
If
$j=0$
the roles of
$\alpha$
and
$\beta$
are reversed. It is proved that the set
${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$
has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.