Minkowski's Question Mark Function

The function  defined by Minkowski for the purpose of mapping the quadratic surds in the open interval  into the rational numbers of  in a continuous, order-preserving manner.  takes a number having continued fraction  to the number

(1)

The function satisfies the following properties (Salem 1943).

1.  is strictly increasing.

2. If  is rational, then  is of the form , with  and  integers.

3. If  is a quadratic surd, then the continued fraction is periodic, and hence  is rational.

4. The function is purely singular (Denjoy 1938).

 can also be constructed as

(2)

where  and  are two consecutive irreducible fractions from the Farey sequence. At the th stage of this definition,  is defined for  values of , and the ordinates corresponding to these values are  for , 1, ...,  (Salem 1943).

The function satisfies the identity

(3)

A few special values include

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

where  is the golden ratio.

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 237-238, 2007.

Conway, J. H. "Contorted Fractions." On Numbers and Games, 2nd ed. Wellesley, MA: A K Peters, pp. 82-86 (1st ed.), 2000.

Denjoy, A. "Sur une fonction réelle de Minkowski." J. Math. Pures Appl. 17, 105-155, 1938.

Girgensohn, R. "Constructing Singular Functions via Farey Fractions." J. Math. Anal. Appl. 203, 127-141, 1996.

Kinney, J. R. "Note on a Singular Function of Minkowski." Proc. Amer. Math. Soc. 11, 788-794, 1960.

Minkowski, H. "Zur Geometrie der Zahlen." In Gesammelte Abhandlungen, Vol. 2. New York: Chelsea, pp. 44-52, 1991.

Salem, R. "On Some Singular Monotone Functions which Are Strictly Increasing." Trans. Amer. Math. Soc. 53, 427-439, 1943.

Tichy, R. and Uitz, J. "An Extension of Minkowski's Singular Functions." Appl. Math. Lett. 8, 39-46, 1995.

Viader, P.; Paradis, J.; and Bibiloni, L. "A New Light on Minkowski's  Function." J. Number Th. 73, 212-227, 1998.

Yakubovich, S. "The Affirmative Solution to Salem's Problem Revisited." 31 Dec 2014. https://arxiv.org/abs/1501.00141.