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Adelic Rogers' formula and an application

Mahbub Alam, Andreas Strömbergsson

Abstract

Recently, Seungki Kim proved an extension of Rogers' mean value formula to the adeles of an arbitrary number field. In this paper we give a new proof Kim's formula, and give a criterion ensuring convergence in this formula. We also discuss one application, namely diophantine approximation over imaginary quadratic number fields with congruence conditions, where we prove an analogue of a famous counting result of W. M. Schimdt.

Adelic Rogers' formula and an application

Abstract

Recently, Seungki Kim proved an extension of Rogers' mean value formula to the adeles of an arbitrary number field. In this paper we give a new proof Kim's formula, and give a criterion ensuring convergence in this formula. We also discuss one application, namely diophantine approximation over imaginary quadratic number fields with congruence conditions, where we prove an analogue of a famous counting result of W. M. Schimdt.
Paper Structure (10 sections, 14 theorems, 107 equations)

This paper contains 10 sections, 14 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Adelic Rogers' formula
  3. Diophantine approximation on $\mathbb{C}$ with congruence conditions
  4. Notation
  5. New proof of Adelic Rogers' formula
  6. Diophantine approximation with congruence conditions
  7. Diophantine approximation over $\mathbb{C}$ with congruence conditions
  8. Analogue of Schmidt schmidt60a
  9. Decomposition of the Haar measure of $\mathbf{GL}_{d}^{1}(\mathbb{C})$
  10. Proof of \ref{['thm:cdiocnew']}

Key Result

Theorem 1.1

Let $1\leq k<d$, and let $f : \mathbf{M}_{d,k}(\mathbb{A}_F)\to\mathbb{R}_{\geq0}$ be a Borel measurable function. Then where the sum over $D$ is over all $m \times k$ row-reduced echelon forms over $F$ of rank $m$ with no zero columns. The relation in eq:arf is an equality of numbers in $\mathbb{R}_{\geq0}\cup\{+\infty\}$, i.e., either both sides are finite and equal or else both are $+\infty$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem : leveque52sullivan82ly16
  • Theorem 1.6
  • Theorem 1.7: alamghoshyu21, Theorem 1.1
  • Lemma 2.1
  • proof
  • ...and 17 more