Details

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.

References

[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.