Density of Rational Points Near Flat/Rough Hypersurfaces

For $n\ge 3$, let $\mathcal{M}\subseteq {\mathbb{R}}^n$ 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 ${\mathrm{N}}_{\mathcal{M}}(\delta ,Q)$ of rationals $\mathbf{a}\mathrm{/}q$, with denominator $q\in [Q,2Q)$ and $\mathbf{a}\in {\mathbb{Z}}^{n-1}$, lying at a distance at most $\delta \mathrm{/}q$ from $\mathcal{M}$. This manuscript provides essentially sharp estimates for ${\mathrm{N}}_{\mathcal{M}}(\delta ,Q)$ throughout the range $\delta \in (Q^{\epsilon -1},1\mathrm{/}2)$ for $d>1+\frac{1}{2n-3}$. 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 $(δ/Q)^{(n-1)/d}Q^n$ (stemming from a so-called Knapp cap), arising in addition to the usual probabilistic term $δQ^n$; the sum of these terms determines the size of $\mathrm{N}_{\mathscr{M}}(δ,Q)$ for $δ\in(Q^{\varepsilon-1},1/2)$. 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.