집중강연

Khintchine’s theorem on manifolds: dynamics, geometry of numbers, and Fourier analysis

 • 발표자 : 김우연

• 소속 : 고등과학원 (허준이펠로우)

• 일시 : 2026년 1월 5일(월)~2026년 1월 9일(금) 14:00~16:00

• 장소 : 28동 102호

Abstract: In this series of talks, we review recent advances in metric Diophantine approximation, focusing on the resolution of Khintchine’s theorem for arbitrary nondegenerate submanifolds of R^n, a longstanding open problem in the field. A key aspect of this problem is its deep connection with the quantitative study of rational points near manifolds.

The foundational breakthrough was achieved by Beresnevich and Yang, who introduced a new strategy combining geometric and dynamical techniques. Their proof involves decomposing a given manifold into ‘generic' and ‘special' parts and utilizing quantitative non-divergence estimates on the space of lattices, together with tools from Minkowski’s geometry of numbers, to handle these components.

More recently, Schindler, Srivastava, and Technau developed a different approach to the associated counting problem. By supplementing quantitative non-divergence with Fourier-analytic methods, they obtained sharper bounds, thereby improving upon the counting estimates established by Beresnevich and Yang.

In these talks, we will present the necessary background in homogeneous dynamics, geometry of numbers, and Fourier analysis, and outline the main ideas behind both the geometric-dynamical proof of Beresnevich and Yang and the Fourier-analytic approach of Schindler, Srivastava, and Technau.