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An effective criterion for periodicity of l-adic continued fractions

Laura Capuano, Francesco Veneziano, Umberto Zannier

Abstract

The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the l-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion, moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol).

An effective criterion for periodicity of l-adic continued fractions

Abstract

The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the l-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion, moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol).

Paper Structure

This paper contains 22 sections, 21 theorems, 93 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. General properties of the continued fractions in Ql
  3. Convergence in Ql
  4. On the convergence in R
  5. Upper bounds for an,pn,qn
  6. Examples
  7. Rationals and the terminating case: Theorem \ref{['T.terminating']}
  8. Proof of part (i) of Theorem \ref{['T.terminating']}
  9. Proof of part (ii) of Theorem \ref{['T.terminating']}
  10. Finiteness of the expansion for varying l
  11. About quadratic expansions
  12. Useful bounds
  13. Convergence in R of the Ruban continued fraction expansion of quadratic irrationals
  14. Purely periodic expansions
  15. Finiteness of purely periodic expansions with a given shape
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\ell$ be a prime number and $\alpha\in{\mathbb Q}$ be a rational number.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 45 more