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 ${\delta }_1\times \dots \times {\delta }_R$ neighborhood of a compact submanifold $M$ of codimension R in $R^M$. We establish an upper bound for this counting function which holds when $M$ satisfies a strong curvature condition, first introduced by Schindler-Yamagishi in [22]. Further, even in the isotropic case when ${\delta }_1=\dots ={\delta }_R=\delta$, we obtain an asymptotic formula which holds beyond the range of distance to $M$ established in [22]. Our result is also a generalization of the work of J.J. Huang [9] 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 $M$ satisfying the strong curvature condition, which agrees with the lower bound obtained by Allen-Wang in [2]. Moreover, for $R>1$, we obtain a new upper bound for the number of rational points on $M$, which goes beyond the bound in an analogue of Serre's dimension growth conjecture for submanifolds of $R^M$.