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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 thequadratic surds in theopen interval
into the rational numbers of
in a continuous, order-preserving manner.
takes a number havingcontinued fraction![x=[0;a_1,a_2,a_3,...]](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fMinkowskisQuestionMarkFunction%2fInline5.svg&f=jpg&w=240)
to the number
 | (1) |
It is implemented in theWolfram LanguageasMinkowskiQuestionMark[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 aquadratic 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 theFarey 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 thegolden ratio.
There are fourfixed points (mod 1) of
, namely
, 1/2,
and
, where
is theMinkowski-Bower constant (Finch 2003, pp. 441-443)
(OEISA048819).
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,FareySequence,Minkowski-Bower ConstantExplore with Wolfram|Alpha
More things to try:
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 inMathematical 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." InGesammelte 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. SequenceA048819 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.http://arxiv.org/abs/1501.00141.Referenced on Wolfram|Alpha
Minkowski's Question Mark FunctionCite this as:
Weisstein, Eric W. "Minkowski's Question MarkFunction." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html