An effective criterion for periodicity of l-adic continued fractions

TL;DR

This paper investigates the periodicity of l-adic continued fractions for rationals and quadratic irrationals, providing explicit criteria to determine when these expansions are periodic, highlighting differences from the real case.

Contribution

It introduces a new explicit criterion for periodicity of l-adic continued fractions for rationals and quadratic irrationals, extending previous qualitative results.

Findings

Rational numbers can have non-terminating periodic expansions in l-adic continued fractions.

Quadratic irrationals do not follow Lagrange's theorem in the l-adic setting.

Explicit criteria for periodicity are established for both rational and quadratic cases.

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).