# Restricted partial quotients

In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction *x* is said to be *restricted*, or composed of **restricted partial quotients**, if the sequence of denominators of its partial quotients is bounded; that is

- [math]\displaystyle{ x = [a_0;a_1,a_2,\dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \ddots}}}} = a_0 + \underset{i=1}{\overset{\infty}{K}} \frac{1}{a_i},\, }[/math]

and there is some positive integer *M* such that all the (integral) partial denominators *a _{i}* are less than or equal to

*M*.

^{[1]}

^{[2]}

## Periodic continued fractions

A regular periodic continued fraction consists of a finite initial block of partial denominators followed by a repeating block; if

- [math]\displaystyle{ \zeta = [a_0;a_1,a_2,\dots,a_k,\overline{a_{k+1},a_{k+2},\dots,a_{k+m}}],\, }[/math]

then ζ is a quadratic irrational number, and its representation as a regular continued fraction is periodic. Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of *a*_{0} through *a*_{k+m}. Historically, mathematicians studied periodic continued fractions before considering the more general concept of restricted partial quotients.

## Restricted CFs and the Cantor set

The Cantor set is a set *C* of measure zero from which a complete interval of real numbers can be constructed by simple addition – that is, any real number from the interval can be expressed as the sum of exactly two elements of the set *C*. The usual proof of the existence of the Cantor set is based on the idea of punching a "hole" in the middle of an interval, then punching holes in the remaining sub-intervals, and repeating this process *ad infinitum*.

The process of adding one more partial quotient to a finite continued fraction is in many ways analogous to this process of "punching a hole" in an interval of real numbers. The size of the "hole" is inversely proportional to the next partial denominator chosen – if the next partial denominator is 1, the gap between successive convergents is maximized.
To make the following theorems precise we will consider CF(*M*), the set of restricted continued fractions whose values lie in the open interval (0, 1) and whose partial denominators are bounded by a positive integer *M* – that is,

- [math]\displaystyle{ \mathrm{CF}(M) = \{[0;a_1,a_2,a_3,\dots]: 1 \leq a_i \leq M \}.\, }[/math]

By making an argument parallel to the one used to construct the Cantor set two interesting results can be obtained.

- If
*M*≥ 4, then any real number in an interval can be constructed as the sum of two elements from CF(*M*), where the interval is given by

- [math]\displaystyle{ (2\times[0;\overline{M,1}], 2\times[0;\overline{1,M}]) = \left(\frac{1}{M} \left[\sqrt{M^2 + 4M} - M \right], \sqrt{M^2 + 4M} - M \right). }[/math]

- A simple argument shows that [math]\displaystyle{ {\scriptstyle[0;\overline{1,M}]-[0;\overline{M,1}]\ge\frac{1}{2}} }[/math] holds when
*M*≥ 4, and this in turn implies that if*M*≥ 4, every real number can be represented in the form*n*+ CF_{1}+ CF_{2}, where*n*is an integer, and CF_{1}and CF_{2}are elements of CF(*M*).^{[3]}

## Zaremba's conjecture

Zaremba has conjectured the existence of an absolute constant *A*, such that the rationals with partial quotients restricted by *A* contain at least one for every (positive integer) denominator. The choice *A* = 5 is compatible with the numerical evidence.^{[4]} Further conjectures reduce that value, in the case of all sufficiently large denominators.^{[5]} Jean Bourgain and Alex Kontorovich have shown that *A* can be chosen so that the conclusion holds for a set of denominators of density 1.^{[6]}

## See also

## References

- ↑ Rockett, Andrew M.; Szüsz, Peter (1992).
*Continued Fractions*. World Scientific. ISBN 981-02-1052-3. https://archive.org/details/continuedfractio0000rock. - ↑ For a fuller explanation of the K notation used here, please see this article.
- ↑ Hall, Marshall (October 1947). "On the Sum and Product of Continued Fractions".
*The Annals of Mathematics***48**(4): 966–993. doi:10.2307/1969389. - ↑ Cristian S. Calude; Elena Calude; M. J. Dinneen (29 November 2004).
*Developments in Language Theory: 8th International Conference, DLT 2004, Auckland, New Zealand, December 13-17, Proceedings*. Springer. p. 180. ISBN 978-3-540-24014-3. https://books.google.com/books?id=z_-SzxRaZ4sC&pg=PA180. - ↑ Hee Oh; Emmanuel Breuillard (17 February 2014).
*Thin Groups and Superstrong Approximation*. Cambridge University Press. p. 15. ISBN 978-1-107-03685-7. https://books.google.com/books?id=XsKfAgAAQBAJ&pg=PA15. - ↑ Bourgain, Jean; Kontorovich, Alex (2014). "On Zaremba's conjecture".
*Annals of Mathematics***180**(1): 137-196. doi:10.4007/annals.2014.180.1.3.

Original source: https://en.wikipedia.org/wiki/ Restricted partial quotients.
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