Real convergence and periodicity of $p$-adic continued fractions

TL;DR

This paper explores the relationship between $p$-adic continued fractions and real quadratic irrationals, establishing necessary conditions for periodicity and proposing probabilistic methods to analyze non-periodicity.

Contribution

It proves that real convergence is necessary for periodicity of $p$-adic continued fractions of quadratic irrationals and develops a probabilistic approach to study their non-periodicity.

Findings

Convergence in $ eal$ is necessary for periodicity.

Supported conjectures on the converse with experimental data.

Probabilistic argument for non-periodicity of Browkin's $p$-adic continued fractions.

Abstract

Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic $p$--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in $\mathbb{R}$ is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in $\mathbb{Q}_p$. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's $p$-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the $p$-adic digits of a quadratic irrational, that holds for almost all $p$-adic numbers.