Alex Smith : Diophantine approximation for hypersurfaces

Posted by Frederick Saia , part of the Number Theory Seminar.

At

Oct. 24, 2025, noon

In

636 SEO

Abstract

Among the nondegenerate C^4 hypersurfaces, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. For all other hypersurfaces, we give a heuristically sharp lower bound for the number of rational points near them, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, involving the application of Ratner's theorems for unipotent orbits, and we will show how our work relates to the dynamical resolution of the Oppenheim conjecture by Margulis.