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 ₁ 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 schindler2022density. Further, even in the isotropic case when ₁==R=, we obtain an asymptotic formula which holds beyond the range of distance to M established in schindler2022density. Our result is also a generalization of the work of J. J. Huang 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 M satisfying the strong curvature condition, which agrees with the lower bound obtained by Allen-Wang in allen2022note. 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 RM.