Density of Rational Points Near Flat/Rough Hypersurfaces

TL;DR

This paper provides sharp estimates for counting rational points near certain hypersurfaces with varying curvature and smoothness, advancing the understanding of Diophantine approximation on rough and flat hypersurfaces.

Contribution

It offers the first sharp bounds for rational points near hypersurfaces with vanishing Gaussian curvature and low regularity, extending previous results.

Findings

Derived estimates for $ ext{N}_ ext{M}( ext{delta},Q)$ across a broad range of $ ext{delta}$.

Identified a geometric term from Knapp caps influencing point counts.

Extended results to the metric theory of Diophantine approximation on rough hypersurfaces.

Abstract

For $n\geq 3$, let $\mathscr{M} \subseteq\mathbb{R}^{n}$ be a compact hypersurface, parametrized by a homogeneous function of degree $d\in \mathbb{R}_{>1}$, with non-vanishing curvature away from the origin. Consider the number $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ of rationals $\mathbf{a}/q$, with denominator $q\in [Q,2Q)$ and $\mathbf{a} \in \mathbb{Z}^{n-1}$, lying at a distance at most $\delta/q$ from $\mathscr{M}$. This manuscript provides essentially sharp estimates for $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ throughout the range $\delta \in (Q^{\varepsilon-1},1/2)$ for $d>1+\tfrac{1}{2n-3}$. Our result is a first of its kind for hypersurfaces with vanishing Gaussian curvature ($d>2$) and those which are rough (meaning not even $C^2$ at the origin which happens when $d<2$). An interesting outcome of our investigation is the understanding of a `geometric' term $(\delta/Q)^{(n-1)/d}Q^n$ (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term $\delta Q^n$; the sum of these terms determines the size of $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ for $\delta\in(Q^{\varepsilon-1},1/2)$. Consequences of our result concern the metric theory of Diophantine approximation on `rough' hypersurfaces -- going beyond the recent break-through of Beresnevich and L. Yang. Further, we establish smooth extensions of Serre's dimension growth conjecture.