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Simultaneous Diophantine approximation with a divisibility condition

Bernard de Mathan

Abstract

In a previous paper, we studied certain sequences of simultaneous rational approximations in ${\bf R}^2$ which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations. Here we show that such results also hold when we add divisibility conditions.

Simultaneous Diophantine approximation with a divisibility condition

Abstract

In a previous paper, we studied certain sequences of simultaneous rational approximations in which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations. Here we show that such results also hold when we add divisibility conditions.
Paper Structure (5 sections, 10 theorems, 174 equations)

This paper contains 5 sections, 10 theorems, 174 equations.

Table of Contents

  1. Introduction.
  2. Proof of Theorem 1.3
  3. Proof of Theorem 1.5
  4. Conclusion.
  5. Acknowledgments

Key Result

Proposition 1.1

Let $(\alpha,\beta)$ be a pair of real numbers. If $(\alpha,\beta)$ satisfies (Bad2), then there exists a sequence of triples of integers $(q_n,r_n,s_n)_{n\ge0}$, with $q_n>0$, satisfying the following conditions: there exists a positive integer $\chi$ and a constant $K>1$ such that Conversely, if there exists a sequence of triples of integers $(q_n,r_n,s_n)$ with $q_n$ positive and unbounded, s

Theorems & Definitions (10)

  • Proposition 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2