Random Walks Based on Continued Fraction Expansions of Real Numbers

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The sequence of digits in the simple continued fraction expansion of and many other famous irrational numbers is conjectured to behave randomly in an appropriate sense. This Demonstration illustrates this behavior by constructing walks based on the continued fraction digits of a number and comparing these walks to similar true random walks.

Contributed by: Alex Jin, Efstathios Konstantinos Chrontsios Garitsis and A. J. Hildebrand(August 2022)
(Based on an undergraduate research project at the Illinois Geometry Lab in Spring 2022.)
Open content licensed under CC BY-NC-SA



This Demonstration was inspired by [1], on the construction of two-dimensional walks from continued fraction expansions of selected real numbers. Given a real number and a base , the walk begins at the origin with the step of length 1 in the direction , where is the digit in the base expansion of .

To construct analogous walks based on the sequence of continued fraction digits of (i.e. the partial quotients in the continued fraction expansion of ), we fix a modulus and reduce the continued fraction digits modulo to obtain a sequence of reduced digits with values in . This sequence is then mapped to a sequence of steps in a walk by mapping a reduced digit to a step in the direction of length . Here is an appropriate normalizing factor since, unlike the digits in the usual base expansion of a typical real number , the reduced continued fraction digits do not have uniform probabilities. Specifically, a positive integer appears in the continued fraction expansion of a typical real number with frequency given by the Gauss–Kuzmin distribution (see, for example, [2, Section 3.4] or [3, Section 15])


Therefore, the proportion of continued fraction digits that are congruent to modulo is given by


Choosing steps of length ensures that the resulting walk has no drift.

The case is mapped to a one-dimensional walk with steps and , where and . This walk can be visualized using ListLinePlot to show the position of the walk after steps, as a function of . When , we obtain a two-dimensional walk in the plane, with steps given as described above.

While the continued fraction expansion of and other famous mathematical constants is conjectured to behave like that of a typical real number , there are some notable exceptions. In particular, quadratic irrationals have a periodic continued fraction expansion, and certain transcendental numbers have an expansion that follows a predictable pattern. For example, the continued fraction expansion of is (see, for example, [2, Appendix A.1]). This regularity is reflected in the continued fraction digit walk based on : the walk has a noticeable drift and is clearly nonrandom. By contrast, walks based on and most other non-quadratic irrationals behave like true random walks constructed from a sequence of independent random digits following the Gauss–Kuzmin distribution.

The controls provide the following options:

1. The "steps" setter bar sets the number of steps of the walk.

2. Select "show axes" to be see the axes, "fixed scale" to use a fixed scale when animating the walk and "colorize" (for moduli ) to display the walk using a rainbow color scheme, with the colors corresponding to the relative position along the walk.

3. Use the "number" setter bar to select from a variety of choices for the number on which the walk is based.

4. Check "show true random walk" to compare with a true random walk, constructed from a simulated sequence of independent Gauss–Kuzmin distributed digits). Select the "randomize" button to generate a new randomization of the true random walk.

5. Use the "modulus" setter bar to set the number of choices for each step in the walk.


[1] F. J. A. Artacho, D. H. Bailey, J. M. Borwein and P. B. Borwein, "Walking on Real Numbers," The Mathematical Intelligencer, 35(1), 2013 pp. 42–60. doi:10.1007/s00283-012-9340-x.

[2] J. Borwein, A. van der Poorten, J. Shallit and W. Zudilin, Neverending Fractions: An Introduction to Continued Fractions, Cambridge, UK: Cambridge University Press, 2014.

[3] A. Ya. Khinchin, Continued Fractions (translated from the Russian by Scripta Technica, Inc., English translation edited by H. Eagle.), 3rd ed., Chicago: University of Chicago Press, 1964.

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