Rational points near planar flat curves

TL;DR

The paper addresses counting rational points near planar finite-type curves by studying $N_f(Q,\delta)$, the number of pairs $(a,q)$ with $1\le q\le Q$ and $||q f(a/q)||<\delta$. The main technique reduces the problem to a model case via a Taylor expansion around a point $a_0$ where the curve has type $d$, and combines discrepancy bounds, truncated Poisson summation, stationary-phase analysis, and dual-function methods to obtain the asymptotic $N_f(Q,\delta)=|I|\delta Q^2+O(\delta^{1/2}(\log(1/\delta))Q^{2-\frac{1}{2(d-1)}}+\delta^{1/d}Q^{2-\frac{1}{d}}+Q^{1+\epsilon})$ for $\delta\in(Q^{-1},\tfrac{1}{2})$ and large $Q$. This extends Huang’s results to finite-type curves and informs higher-dimensional analogues under curvature hypotheses. The work advances the understanding of how curvature and degeneracy affect the distribution of rational points near manifolds of various codimensions, with potential applications to Diophantine approximation on manifolds.

Abstract

We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.

Theorems & Definitions (16)

• Definition 1.1
• Theorem 1.2
• Example 1.3
• Example 1.4: Hua20, Example 4
• Remark 1.5
• Theorem 1.6: Hua15, Theorem 3
• Example 1.7
• Conjecture 1.8: Hua20, Conjecture 1
• Theorem 1.9: Hua20, Theorem 1
• Definition 1.10