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This is a Quantitative Study study.

October 12, 2025Open Access

Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

Puntos clave

The study shows effective equidistribution for orbits of unipotent subgroups in rank 2 homogeneous spaces, enhancing understanding in algebraic geometry.
Key evidence includes a polynomial error rate for orbits of unipotent subgroups in almost simple linear algebraic groups of rank 2.
The approach utilizes effective equidistribution theorems applied to the distribution of values of quadratic forms at integer points.
This work suggests substantial advancements in quantitative results related to the Oppenheim Conjecture and its implications for number theory.

Resumen

We establish effective equidistribution theorems, with a polynomial error rate, for orbits of unipotent subgroups in quotients of quasi-split, almost simple Linear algebraic groups of absolute rank 2. As an application, inspired by the results of Eskin, Margulis and Mozes, we establish quantitative results regarding the distribution of values of an indefinite ternary quadratic form at integer points, giving in particular an effective and quantitative proof of the Oppenheim Conjecture.

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