In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated: $$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\frac{(-1)^n}{p_{n-1}-q_{n-1}\alpha}$$

for $n\geq 0$ with $p_{-1}=1$ and $q_{-1}=0$, where $$\alpha=[a_0;a_1,...,a_{j-1},\alpha_j]$$ ($\alpha_j$ being the tail of the continued fraction development of $\alpha$) and, as usual, $$\frac{p_n}{q_n}=[a_0;a_1,...,a_{n}]$$

In the book, the authors explain the name with

It turns out that one may usefully think of $\left|\log(\left|p_{n-1}-q_{n-1}\alpha\right|)\right|$ as measuring the weighted distance that the continued fraction has traversed in moving from $\alpha$ to $\alpha_n$

My questions are:

  1. Does this distance formula only hold for quadratic irrationals?
  2. Where can I find additional literature / information on this formula? (I couldn't find any other references for it so far)

1 Answer 1

  1. This formula is valid for any continued fraction. But for quadratic irrationals it is especially useful because it allows to express fundamental unit of corresponding field in terms of continued fraction expansion of $\sqrt{n}$. For a reduced quadratic irrational $\omega=[0;\overline{a_1,\ldots,a_n}]$ with period $n=\mathrm{per}(\omega)$ we write $$ {\mathrm{per}}_e(\omega)=\begin{cases} n, &\text{if $n={\mathrm{per}}(\omega)$ is even;} \\ 2n,&\text{if $n={\mathrm{per}}(\omega)$ is odd.} \end{cases} $$ In this case, the fundamental unit can be found using Smith’s formula $$ \varepsilon_0^{-1}(\omega)=\omega T(\omega)T^2(\omega)\ldots T^{\mathrm{per}_e(\omega)-1}(\omega), $$ where $T(\alpha)$ stands for the Gauss map: $T(\alpha)=\left\{{1}/{\alpha}\right\}$.

  2. The Smiths article (Smith H. J. S. Note on the Theory of the Pellian Equation and of Binary Quadratic Forms of a Positive Determinant. Proc. London Math. Soc., 1875, s1-7, 196-208.) is here.

A good book about this topic is B. A. Venkov, Elementary number theory, ONTI, Moscow 1931; English transl., Wolters-Noordhoff Publ., Groningen 1970. (see zbmath for review, and chapter 2 for continued fractions).

Applications to quadratic irrationals (Gauss-Kuz'min statistica and distribution of lengths) can be found in Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals.

  • $\begingroup$ Thanks a lot, Alexey! This is really helpful for me. Surprising that you mention the fundamental unit, as I stumbled across it in the same context, but haven't associated it with the distance formula (it was in connection with the Regulator of a number field) So good to know & thanks again! $\endgroup$
    – Marcus
    Sep 15, 2021 at 11:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.