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

Runlin Zhang

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.

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

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.
Paper Structure (32 sections, 19 theorems, 201 equations)

This paper contains 32 sections, 19 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Counting
  4. Equidistribution of infinite homogeneous measures
  5. Outline of the paper
  6. Integer points on ternary affine quadrics
  7. Preliminaries
  8. From equidistribution to counting
  9. Holder continuity and effective equidistribution
  10. Boundary values of $f_{\varphi}$
  11. Proof of Theorem \ref{['theorem_equidistribution_infinite_homogeneous_quadric']} assuming Theorem \ref{['theorem_Holder_continuous_quadrics']}
  12. Effective equidistribution: the focusing case
  13. Convention
  14. Model space
  15. Effective equidistribution of lines on tori defined by number fields
  16. ...and 17 more sections

Key Result

Theorem 1.1

For every rational point $x\in X(k)$ and $\varphi \in C_c^{\infty}(K_0 \backslash Y)$ with $\int_Y \varphi(y) \,\mathrm{d}\mathrm{m}_Y(y) =1$, there exist two monic polynomials $p_1,p_2$ of degree $l_1+l_2$ and $l_1+l_2-1$ respectively, $c_1,c_2>0$, $c_2(\varphi) \in \mathbb{R}$ and $\delta\in (0,0.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 18 more