Effective count of integer points on ternary affine quadrics and effective equidistribution

TL;DR

This paper develops effective equidistribution results for counting integer points on ternary affine quadrics over number fields, utilizing mixing and Eisenstein series to improve previous theorems.

Contribution

It introduces new effective counting methods for integer points on affine quadrics over number fields, extending prior work by Oh-Shah and Kelmer-Kontorovich.

Findings

Established effective equidistribution for infinite homogeneous measures.

Derived smooth counting theorems over number fields.

Applied Eisenstein series to obtain authentic counting results.

Abstract

We study the effective equidistribution of certain infinite homogeneous measures and related counting problems through mixing. In this way, we obtain smooth versions of counting theorems studied by Oh-Shah and later by Kelmer-Kontorovich over a number field. In the appendix, we apply the meromorphic continuation of Hilbert-Asai Eisenstein series to obtain the authentic counting.