Counting rational points near manifolds: a refined estimate, a conjecture and a variant
Jonathan Hickman, Rajula Srivastava, James Wright
TL;DR
This work advances the understanding of rational points near manifolds by developing a refined Fourier-analytic framework for intermediate-dimension manifolds under a curvature condition (CC). It introduces a non-isotropic counting approach and a self-improving bootstrapping mechanism, yielding improved exponent bounds and establishing a general refined estimate that surpasses prior results for codimensions $R \ge 3$. The authors formulate a conjecture for codimension $2$ and provide supporting evidence via complex-analytic reformulations (\mathbb{C}-CC) and Gaussian-rational variants, including sharp bounds in many cases and explicit examples demonstrating sharpness in certain regimes. They also extend the analysis to Gaussian rationals, showing sharp, subpolynomial-loss bounds that align with the conjectured codimension-2 behavior for large $n$. Overall, the paper uses a blend of stationary-phase analysis, Poisson summation, and summation-by-parts to push the limits of Fourier-analytic methods in Diophantine approximation on manifolds and to connect real and complex-analytic geometry in this counting problem.
Abstract
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 $n \geq 2$ in which rationals are replaced with Gaussian rationals.