Effective version of Ratner’s equidistribution theorem for SL(3, R)

April 21, 2025

A mathematician from the National University of Singapore (NUS) has introduced a new framework to study the quantitative equidistribution of unipotent orbits in homogeneous spaces and resolve an important case in this area. 

This research focuses on a family of important dynamical systems called unipotent dynamical systems. These dynamical systems are important because they are closely related to several important problems from other branches of mathematics, including number theory, representation theory, and geometry. A fundamental theorem in this field is Ratner’s equidistribution theorem (1991), which establishes that unipotent orbits in these systems become evenly distributed in the long run. However, a natural question is whether the asymptotic result can be proven with a quantitative rate. In simpler terms, it means whether we can determine how quickly a finite orbit spreads out to match its expected distribution. This question has been a major challenge in dynamical systems and is connected to several important problems in number theory, including a quantitative version of the Oppenheim conjecture. However, due to the lack of efficient approaches, there have been very few results in this direction, leaving the problem open for many important cases.

In the present paper, Associate Professor YANG Lei from the Department of Mathematics at NUS has proved a quantitative version of Ratner’s equidistribution theorem for an important case: unipotent orbits in SL(3, R)/SL(3, Z). He has also developed a general framework to study the quantitative behaviour of unipotent orbits. The core techniques include a bootstrapping argument that allows for multi-scale analysis, and a new geometric model that connects the quantitative behaviour of unipotent orbits to the geometric properties of the model. In particular, the geometric model is closely related to another famous and longstanding conjecture from incidence geometry and harmonic analysis called Kakeya’s conjecture.

This work not only advances our understanding of unipotent dynamics but also introduces tools that could have broader applications in other mathematical fields.

Figure illustrates an analogy for the research work: imagine dropping a handful of sand onto a table. Ratner’s theorem predicts that the grains will eventually spread out evenly. This research explains how quickly that happens and whether any clumps remain. [Image generated by AI tool]

 

Reference
Lei Yang*, “Effective version of Ratner’s equidistribution theorem for SL(3, R)” to appear in Annals of Mathematics.