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Minkowski's Question Mark Function
Minkowski's question mark function is 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 ![x=[0;a_1,a_2,a_3,...]](https://mathworld.wolfram.com/images/equations/MinkowskisQuestionMarkFunction/Inline5.svg)
to the number
 |
(1)
|
It is implemented in the Wolfram Language as MinkowskiQuestionMark[x].
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.
There are four fixed points (mod 1) of 
, namely 
, 1/2, 
and 
, where 
is the Minkowski-Bower
constant (Finch 2003, pp. 441-443) 
(OEIS A048819).
Values 
with large terms in their continued fractions cause 
to have a large section of repeating 0's or 9's (E. Pegg,
Jr., pers. comm., Jan. 5, 2023). Some examples include
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
See also
Devil's Staircase, Farey Sequence, Minkowski-Bower ConstantExplore with Wolfram|Alpha
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.Finch, S. R. "Minkowski-Bower Constant." §6.9 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 441-443, 2003.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.Sloane, N. J. A. Sequence A048819 in "The On-Line Encyclopedia of Integer Sequences."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.Referenced on Wolfram|Alpha
Minkowski's Question Mark FunctionCite this as:
Weisstein, Eric W. "Minkowski's Question Mark Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html