Lattice points in thickened parabolas and rational points near hypersurfaces

TL;DR

This paper characterizes rational quadrics among hypersurfaces in R^n as those least well approximated by rational points, providing sharp bounds for rational points near other hypersurfaces using dynamical methods.

Contribution

It introduces a new dynamical approach leveraging Ratner's theorems to analyze rational point distribution near hypersurfaces, improving previous bounds.

Findings

Rational quadrics are uniquely poorly approximated by rational points.

Established sharp lower bounds for rational points near non-quadratic hypersurfaces.

Applied Ratner's theorems to study algebraic subgroups of SL_n(Q).

Abstract

Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M other than a rational quadric, we prove a heuristically sharp lower bound for the number of rational points very near M, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, and rely on an application of Ratner's theorems to 1-parameter unipotent subgroups U of SL_n(R) such that u - Id has rank at most 2 for every u in U. As part of our work, we study the algebraic subgroups of SL_n(Q) whose collection of real points can contain such a subgroup.