Counting rational points near manifolds: a refined estimate, a conjecture and a variant

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of ${\mathbb{R}}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$ case we formulate a conjecture concerning this count. The conjecture is motivated in part by interpreting certain codimension $2$ submanifolds of ${\mathbb{R}}^{2m+2}$ as complex hypersurfaces in ${\mathbb{C}}^{m+1}$ and using the complex structure to provide a natural reformulation of the curvature condition. Finally, we provide further evidence for the conjecture by proving a natural variant for $n\ge 2$ in which rationals are replaced with Gaussian rationals.