3
$\begingroup$

A friend and I were discussing the properties of continued fractions (as "best" approximations). For fun, we checked the continued fractions of Liouville's constant. The terms in the sequence fit a very clear pattern, see

http://mathworld.wolfram.com/LiouvillesConstant.html

Is there an actual reference to show that the sequence is a nested sequence? Here are more specific questions that focus on certain issues. Let $x = \sum_{i=1}^{\infty} a^{-i!}$. Is there a reference for

  1. The only terms in the continued fraction are $$ 1, a-2, a, a+1 \textrm{ and } a^k-1 \textrm{ for some } k \geq 1. $$

  2. The $n^\textrm{th}$ incrementally largest term (considering only those entirely of the form $a^k-1$) occurs precisely at position $2^n-1$.

  3. The exponent $k$ in the largest term at position $2^n-1$ is $n!\cdot (n-1)$.

  4. What is the algorithm to build the sequence as a nested sequence?

$\endgroup$
7
$\begingroup$

This was first done in my paper, Simple Continued Fractions for Some Irrational Numbers II, published in J. Number Theory 14 (1982), 228-231.

From the proof there you can deduce the 4 things you listed.

$\endgroup$
  • 2
    $\begingroup$ It's a shame your paper is not the references of the webpage! It is also absent from the references in oeis.org/A058304 ... Many thanks! $\endgroup$ – ARG Apr 30 '17 at 10:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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