  In this manuscript, we initiate the study of the number of rational points
with bounded denominators, contained in a non-isotropic
$\delta_1\times\ldots\times \delta_R$ neighborhood of a compact submanifold
$\mathcal{M}$ of codimension $R$ in $\mathbb{R}^{M}$. We establish an upper
bound for this counting function which holds when $\mathcal{M}$ satisfies a
strong curvature condition, first introduced by Schindler-Yamagishi in
\cite{schindler2022density}. Further, even in the isotropic case when
$\delta_1=\ldots=\delta_R=\delta$, we obtain an asymptotic formula which holds
beyond the range of distance to $\mathcal{M}$ established in
\cite{schindler2022density}. Our result is also a generalization of the work of
J.J. Huang \cite{huangduke} for hypersurfaces.
  As an application, we establish for the first time an upper bound for the
Hausdorff dimension of the set of weighted simultaneously well approximable
points on a manifold $\mathcal{M}$ satisfying the strong curvature condition,
which agrees with the lower bound obtained by Allen-Wang in
\cite{allen2022note}. Moreover, for $R>1$, we obtain a new upper bound for the
number of rational points \textit{on} $\mathcal{M}$, which goes beyond the
bound in an analogue of Serre's dimension growth conjecture for submanifolds of
$\mathbb{R}^M$ .
