Question mark function

In mathematics, the question mark function is a function denoted ?(x) whose definition is due to Hermann Minkowski. If <math>[a_0; a_1, a_2, \cdots]<math> is the continued fraction representation of an irrational number x, then

<math>{\rm ?}(x) = a_0 + 2 \sum_{n=1}^\infty (-1)^{n+1}2^{-a_1 – \cdots -a_n}<math>

whereas if <math>[a_0; a_1, a_2, \cdots a_m]<math> is a continued fraction for a rational number, then

<math>{\rm ?}(x) = a_0 + 2 \sum_{n=1}^m (-1)^{n+1}2^{-a_1 – \cdots -a_n}<math>

It should be noted that if a_m is greater than one, then <math>[a_0; a_1, a_2, \cdots a_m-1, 1]<math> is also a continued fraction for the same number, but the two expressions give identical values for ?(x).

For rational numbers the function may also be defined recursively; if p/q and r/s are reduced fractions such that |ps – rq| = 1 (so that they are adjacent elements of a row of the Farey sequence) then

<math>?(\frac{p+r}{q+s}) = \frac12 (?(\frac{p}{q}) + ?(\frac{r}{s}))<math>

Note that in general, any matrix <math>\begin{pmatrix} p & q \\ r & s \end{pmatrix}<math> having unit determinant is by definition a member of the modular group. The question mark function is self-similar, with the modular group describing the self-similarity. This follows in part because if <math>p_{n-1}/q_{n-1}<math> and <math>p_{n}/q_{n}<math> are two successive convergents of a continued fraction, then the matrix <math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}<math> has determinant plus or minus 1.

Properties of ?(x)

The question mark function is a strictly increasing and continuous function. It has a derivative almost everywhere, of value zero. At other points either the derivative is not defined at all, or exists in the sense of being infinite--that is, such that (?^-1)'(?(x)) equals zero. It sends rational numbers to dyadic rational numbers, meaning those whose base two representation terminates. It sends quadratic irrationalities to non-dyadic rational numbers. If ?(x) is irrational, then x is either algebraic of degree greater than two, or transcendental. The function is invertible, and the inverse function has also attracted the attention of various mathematicians, in particular John Conway, whose notation for ?^-1(x) is x with a box drawn around it. It is an odd function, and satisfies the functional equation ?(x+1) = ?(x)+1; consequently ?(x)-x is an odd periodic function with period one.

The Minkowski question mark function is a special case of fractal curves known as de Rham curves.

References

• Biblioni, L., Paradis, J., Viader, P., A New Light on Minkowski's ?(x) Function, Journal of Number Theory, 73 (1998), 212–227 http://www.econ.upf.es/eng/research/onepaper.php?id=226

• Biblioni, L., Paradis, J., Viader, P., The Derivative of Minkowski's Singular Function, Journal of Mathematical Analysis and Applications, (2001) 107–125 http://www.econ.upf.es/eng/research/onepaper.php?id=378

• Conley, Randolph M. A Survey of the Minkowski ?(x) Function, Master's Thesis, West Virginia University, 2003 http://kitkat.wvu.edu:8080/files/3055.1.Conley_Randolph_thesis.pdf

• Vepstas, Linas, The Minkowski Question Mark and the Modular Group SL(2,Z) 2004, http://www.linas.org/math/chap-minkowski/chap-minkowski.html

• Vepstas, Linas, Modular Fractal Measures 2004, http://www.linas.org/math/fdist/fdist.html conjectures that the derivative of the question mark is the limit of a modular form.