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.
Categories: math.NT math.CA