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Minkowski's Question Mark Function

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MinkowskiQuestionMark

Minkowski's question mark function is the function y=?(x) defined by Minkowski for the purpose of mapping the quadratic surds in the open interval (0,1) into the rational numbers of (0,1) in a continuous, order-preserving manner. ?(x) takes a number having continued fraction x=[0;a_1,a_2,a_3,...] to the number

?(x)=sum_(k)((-1)^(k-1))/(2^((a_1+...+a_k)-1)).
(1)

It is implemented in the Wolfram Language as MinkowskiQuestionMark[x].

The function satisfies the following properties (Salem 1943).

1. ?(x) is strictly increasing.

2. If x is rational, then ?(x) is of the form k/2^s, with k and s integers.

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

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

?(x) can also be constructed as

?((p+p^')/(q+q^'))=(?(p/q)+?(p^'/q^'))/2,
(2)

where p/q and p^'/q^' are two consecutive irreducible fractions from the Farey sequence. At the nth stage of this definition, ?(x) is defined for 2^n+1 values of x, and the ordinates corresponding to these values are x=k/2^n for k=0, 1, ..., 2^n (Salem 1943).

The function satisfies the identity

?(1/(k^n))=1/(2^(k^n-1)).
(3)

A few special values include

?(0)=0
(4)
?(1/3)=1/4
(5)
?(1/2)=1/2
(6)
?(phi-1)=2/3
(7)
?(2/3)=3/4
(8)
?(1/2sqrt(2))=4/5
(9)
?(1/2sqrt(3))=(84)/(85)
(10)
?(1)=1,
(11)

where phi is the golden ratio.

There are four fixed points (mod 1) of ?(x), namely x=0, 1/2, f and 1-f, where f<1/2 is the Minkowski-Bower constant (Finch 2003, pp. 441-443) f=0.42037... (OEIS A048819).

Values x with large terms in their continued fractions cause ?(x) to have a large section of repeating 0's or 9's (E. Pegg, Jr., pers. comm., Jan. 5, 2023). Some examples include

?(6^(1/3))=1.9530189847656249...9_()_(142)6...
(12)
?(20^(1/3))=2.81250...0_()_(43)2...
(13)
?(pi)=3.1562476158142089843749...9_()_(72)8....
(14)

See also

Devil's Staircase, Farey Sequence, Minkowski-Bower Constant

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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 ?(x) Function." J. Number Th. 73, 212-227, 1998.Yakubovich, S. "The Affirmative Solution to Salem's Problem Revisited." 31 Dec 2014. http://arxiv.org/abs/1501.00141.

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Minkowski's Question Mark Function

Cite this as:

Weisstein, Eric W. "Minkowski's Question Mark Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html

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