Minkowski,s Question Mark Function
المؤلف:
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H
المصدر:
Experimental Mathematics in Action. Wellesley, MA: A K Peters
الجزء والصفحة:
...
8-5-2020
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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 ![x=[0;a_1,a_2,a_3,...]](https://mathworld.wolfram.com/images/equations/MinkowskisQuestionMarkFunction/Inline5.gif)
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
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.
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