Density of Rational Points Near Flat/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.