Generalization of continued fraction to field $\mathbb{Q}_p$

Question

Any reference for generalization or extension of simple continued fraction to field $\mathbb{Q}_p$?

Answer 1

There is no single extension of "simple continued fractions" to $p$-adic numbers $\mathbb{Q}_p$ which shares all the properties of the real simple continued fraction that proceeds by a division algorithm.

An illustration of how the direct generalization breaks down is given in the answer to continued fraction expression for $\sqrt{2}$ in $\mathbb{Q}_7$.

We should add to this an example addressing the Comment by @reuns, the simplest of simple continued fractions:

$$ \phi = 1 + 1/(1 + 1/(1 + \ldots)) $$

which is the real continued fraction expansion of the golden ratio, does not converge in the $p$-adic field $\mathbb{Q}_p$ (because its convergents are not a Cauchy sequence there).

It is well-known that the convergents of this particular continued fraction are ratios of consecutive Fibonacci numbers:

$$ \frac{h_n}{k_n} = \frac{F_{n+1}}{F_n} $$

Thus the difference of consecutive convergents simplifies by Cassini's identity:

$$ \begin{align*} \frac{F_{n+2}}{F_{n+1}} - \frac{F_{n+1}}{F_n} &= \frac{F_{n+2}F_n - F_{n+1}^2}{F_n F_{n+1}} \\ &= \frac{(-1)^{n+1}}{F_n F_{n+1}} \end{align*} $$

This shows the sequence of convergents is not Cauchy in the $p$-adic metric, which has two terms close accordingly as their difference is divisible by a higher power of $p$. Here the numerator is $1$ (in absolute value), so that consecutive convergents are at least one unit apart (farther as the denominator may be divisible by a power of $p$).

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A readable introduction to $p$-adic continued fractions is an undergraduate research paper by Matthew Moore (2006). The abstract of that paper begins:

Simple continued fractions in $\mathbb{R}$ have a single definition and algorithms for calculating them are well known. There also exists a well known result which states that $\sqrt{m}$, $m \in \mathbb{Q}$ and $m \ge 0$, always has a periodic continued fraction representation. In $\mathbb{Q}_p$, the field of $p$-adics, however, no single algorithm exists which always produces a periodic continued fraction for $\sqrt{m}$, and no result is available to guarantee the existence of one.

On the other hand it is possible to define "simple continued fraction" in ways that allow us to construct one that converges for any $\alpha \in \mathbb{Q}_p$. The central difficulty is described by the above paper:

The standard algorithm for computing a continued fraction from a real number relies fundamentally upon the floor function. The floor function is a function such that, when passed a real number, returns the greatest integer less than or equal to that number. The problem with extending the standard algorithm in $\mathbb{R}$ to $\mathbb{Q}_p$ is that it is not clear which part of a $p$-adic number is “decimal” and which part is “integer”. For this reason, it is necessary to define a floor function from scratch. However, due to the ambiguities inherent in this, no clear choice for a floor function in $\mathbb{Q}_p$ exists, although there are several reasonable candidates.

The paper then presents two such algorithms for convergent continued fractions, and explores the connection with being able to characterize rational numbers (which terminate in the real case) and square roots (which have periodic expansions in the real case). References in the paper include citations to papers by Jerzy Browkin on "Continued Fractions in Local Fields", parts I and II.

More references to "various attempts to create a satisfactory continued fraction theory in the $p$-adic setting" are given in the paper p-Adic Continued Fractions of Liouville Type by Laohakosal and Ubolsri (1987).