Counting Rational Points In Non-Isotropic Neighborhoods of Manifolds

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 $\mathbb{R}^M$ .