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 δ₁×…× δR neighborhood of a compact submanifold M of codimension R in RM. 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 δ₁=…=δR=δ, 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 .