Real convergence and periodicity of $p$-adic continued fractions
Giuliano Romeo
TL;DR
The paper investigates when Browkin-type p-adic continued fractions are eventually periodic for quadratic irrationals. It proves that periodicity in the p-adic setting forces real convergence to one of the real quadratic conjugates, via a real-analytic transfer grounded in Wall's theorem, and it develops a probabilistic framework under uniform digit distribution to argue that non-periodicity should be typical for these algorithms. The work combines rigorous results with conjectures supported by computations, suggesting that real convergence behavior can characterize periodicity and offering practical indicators for detecting non-periodicity. This bridges p-adic dynamics with real quadratic arithmetic and informs how to assess periodicity through real-convergence diagnostics, with implications for Browkin-type algorithms and their applicability to quadratic irrationals.
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.