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| [[Image:Minkowski question mark.svg|300px|thumb|Minkowski question mark function.]]
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| [[File:Minkowski qn mark fcn.gif|500px|thumb|Left: {{math|?(''x'')}}. Right: {{math|?(''x'') − ''x''}}.]]
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| In [[mathematics]], the '''Minkowski question mark function''' (or the '''slippery [[Singular function|devil's staircase]]'''), denoted by {{math|size=120%|?(''x'')}}, is a [[Function (mathematics)|function]] possessing various unusual [[fractal]] properties, defined by [[Hermann Minkowski]] in 1904. It maps [[quadratic irrational]]s to [[rational number]]s on the [[unit interval]], via an expression relating the [[continued fraction]] expansions of the quadratics to the [[binary expansion]]s of the rationals, given by [[Arnaud Denjoy]] in 1938. In addition, it maps rational numbers to [[dyadic rational]]s, as can be seen by a recursive definition closely related to the [[Stern–Brocot tree]].
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| ==Definition==
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| If <math>[a_0; a_1, a_2, \ldots]</math> is the [[continued fraction|continued fraction representation]] of an [[irrational number]] {{mvar|x}}, then
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| :<math>{\rm ?}(x) = a_0 + 2 \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2^{a_1 + \cdots + a_n}}</math>
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| whereas:
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| If <math>[a_0; a_1, a_2, \ldots, a_m]</math> is a continued fraction representation of a [[rational number]] {{mvar|x}}, then
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| :<math>{\rm ?}(x) = a_0 + 2 \sum_{n=1}^m \frac{(-1)^{n+1}}{2^{a_1 + \cdots + a_n}}</math>
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| ==Intuitive explanation==
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| To get some intuition for the definition above, consider the different ways
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| of interpreting an infinite string of bits beginning with 0 as a real number in {{closed-closed|size=120%|0,1}}.
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| One obvious way to interpret such a string is to place a binary point after the first 0 and read the string as a [[binary numeral system|binary]] expansion: thus, for instance, the string 001001001001001001001001... | |
| represents the binary number 0.010010010010..., or {{math|2/7}}. Another interpretation
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| views a string as the [[continued fraction]] {{math|size=120%|[0; ''a''<sub>1</sub>, ''a''<sub>2</sub>, … ]}}, where the integers {{mvar|size=120%|a<sub>i</sub>}} are the run lengths in a [[run-length encoding]] of the string. The same example string 001001001001001001001001... then
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| corresponds to {{math|size=120%|1=[0; 2, 1, 2, 1, 2, 1, …] = }} {{sfrac|{{sqrt|3}} − 1|2}}. If the string ends in an infinitely long run of the same bit, we ignore it and terminate the representation; this is suggested by the formal "identity":
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| : {{math|size=120%|1=[0; ''a''<sub>1</sub>, … ,''a''<sub>''n''</sub>, ∞] = }} {{math|size=120%|1=[0; ''a''<sub>1</sub>, … ,''a''<sub>''n''</sub>+1/∞] = }} {{math|size=120%|1=[0; ''a''<sub>1</sub>, … ,''a''<sub>''n''</sub>+0] = }} {{math|size=120%|[0; ''a''<sub>1</sub>, … ,''a''<sub>''n''</sub>]}}.
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| The effect of the question mark function on {{closed-closed|size=120%|0,1}} can then be understood as mapping the second interpretation of a string to the first interpretation of the same string,<ref name=Fin4412>Finch (2003) pp.441–442</ref><ref name=PF95/> just as the [[Cantor function]] can be understood as mapping a triadic [[base 3]] representation to a base 2 representation. Our example string gives the equality
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| :<math>?\left(\frac{\sqrt3-1}{2}\right)=\frac{2}{7}.</math>
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| ==Recursive definition for rational arguments==
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| For rational numbers in the unit interval, the function may also be defined [[recursion|recursively]]; if {{math|''p/q''}} and {{math|''r/s''}} are [[reduced fraction]]s such that {{math|1={{!}} ''ps'' − ''rq'' {{!}} = 1}} (so that they are adjacent elements of a row of the [[Farey sequence]]) then<ref name=PF95>Pytheas Fogg (2002) p.95</ref>
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| :<math>?\left(\frac{p+r}{q+s}\right) = \frac12 \left(?\bigg(\frac pq\bigg) + {}?\bigg(\frac rs\bigg)\right)</math>
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| Using the base cases
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| :<math>?\left(\frac{0}{1}\right) = 0 \quad \mbox{ and } \quad ?\left(\frac{1}{1}\right)=1</math>
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| it is then possible to compute {{math|?(''x'')}} for any rational {{mvar|x}}, starting with the [[Farey sequence]] of order 2, then 3, etc.
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| If <math>p_{n-1}/q_{n-1}</math> and <math>p_{n}/q_{n}</math> are two successive convergents of a [[continued fraction]], then the matrix
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| :<math>\begin{pmatrix} p_{n-1} & p_{n} \\ q_{n-1} & q_{n} \end{pmatrix}</math>
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| has [[determinant]] ±1. Such a matrix is an element of <math>S^*L(2,Z)</math>, the group of two-by-two matrices with determinant ±1. This group is related to the [[modular group]].
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| ===Algorithm===
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| This recursive definition naturally lends itself to an [[algorithm]] for computing the function to any desired degree of accuracy for any real number, as the following [[C (programming language)|C]] function demonstrates. The algorithm descends the [[Stern–Brocot tree]] in search of the input {{mvar|x}}, and sums the terms of the binary expansion of <math>y = \; ?(x)</math> on the way. As long as the '''[[loop invariant]]''' <math>qr-ps=1</math> remains satisfied there is no need to reduce the fraction <math>\frac m n = \frac{p+r}{q+s},</math> since it is already in lowest terms. Another invariant is <math>\frac p q \le x < \frac r s.</math> The '''for''' loop in this program may be analyzed somewhat like a '''while''' loop, with the conditional break statements in the first three lines making out the condition. The only statements in the loop that can possibly affect the invariants are in the last two lines, and these can be shown to preserve the truth of both invariants as long as the first three lines have executed successfully without breaking out of the loop. A third invariant for the body of the loop (up to floating point precision) is <math>y \le \; ?(x) < y + d,</math> but since {{mvar|d}} is [[division by two|halved]] at the beginning of the loop before any conditions are tested, our conclusion is only that <math>y \le \; ?(x) < y + 2d</math> at the termination of the loop.
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| To [[Loop variant|prove termination]], it is sufficient to note that the sum <math>q+s</math> increases by at least 1 with every iteration of the loop, and that the loop will terminate when this sum is too large to be represented in the primitive C data type '''long'''. However, in practice, the conditional break when "y+d==y" is what ensures the termination of the loop in a reasonable amount of time.
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| <source lang="c"> | |
| /* Minkowski's question mark function */
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| double minkowski(double x) {
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| long p=x; if ((double)p>x) --p; /* p=floor(x) */
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| long q=1, r=p+1, s=1, m, n;
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| double d=1, y=p;
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| if (x<(double)p||(p<0)^(r<=0)) return x; /* out of range ?(x) =~ x */
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| for (;;) /* invariants: q*r-p*s==1 && (double)p/q <= x && x < (double)r/s */
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| {
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| d/=2; if (y+d==y) break; /* reached max possible precision */
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| m=p+r; if ((m<0)^(p<0)) break; /* sum overflowed */
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| n=q+s; if (n<0) break; /* sum overflowed */
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|
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| if (x<(double)m/n) r=m, s=n;
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| else y+=d, p=m, q=n;
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| }
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| return y+d; /* final round-off */
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| }
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| </source>
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| ==Self-symmetry==
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| The question mark is clearly visually self-similar. A [[monoid]] of self-similarities may be generated by two operators {{mvar|S}} and {{mvar|R}} acting on the unit square and defined as follows:
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| :<math>\begin{array}{lcl}
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| S(x, y) &=& \left( \frac{x}{x+1}, \frac{y}{2} \right) \\
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| R(x, y) &=& \left( 1-x, 1-y \right)\,.
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| \end{array}</math>
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| Visually, {{mvar|S}} shrinks the unit square to its bottom-left quarter, while {{mvar|R}} performs a [[point reflection]] through its center.
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| A point on the [[function graph|graph]] of ? has coordinates {{math|(''x'', ?(''x''))}} for some {{mvar|x}} in the unit interval. Such a point is transformed by {{mvar|S}} and {{mvar|R}} into another point of the graph, because ? satisfies the following identities for all <math>x\in [0,1]</math>:
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| :<math>\begin{array}{lcl}
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| ?\left(\frac{x}{x+1}\right) &=& \frac{?(x)}{2} \\
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| ?(1-x) &=& 1-?(x)\,.
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| \end{array}</math>
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| These two operators may be repeatedly combined, forming a monoid. A general element of the monoid is then
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| :<math>S^{a_1} R S^{a_2} R S^{a_3} \cdots</math>
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| for positive integers <math>a_1, a_2, a_3, \ldots</math>. Each such element describes a self-similarity of the question mark function. This monoid is sometimes called the '''[[period-doubling monoid]]''', and all period-doubling fractal curves have a self-symmetry described by it (the [[de Rham curve]], of which the question mark is a special case, is a category of such curves). Note also that the elements of the monoid are in correspondence with the rationals, by means of the identification of <math>a_1, a_2, a_3, \ldots</math> with the continued fraction <math>[0; a_1, a_2, a_3, \ldots]</math>. Since both
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| :<math>S: x \mapsto \frac{x}{x+1}</math>
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| and
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| :<math>T: x \mapsto 1-x</math>
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| are [[linear fractional transformation]]s with integer coefficients, the monoid may be regarded as a subset of the [[modular group]] PSL(2,'''Z''').
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| ==Properties of ?(''x'')==
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| {{wide image|Minkowski'sQuestionMarkLessTheIdentity.png|1024px|<center>{{bigmath|?(''x'') − ''x''}}</center>}}
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| The question mark function is a [[strictly increasing]] and continuous,<ref name=Fin442>Finch (2003) p.442</ref> but not [[absolutely continuous]] function. The [[derivative]] vanishes on the [[rational number]]s. There are several constructions for a [[measure (mathematics)|measure]] that, when integrated, yields the question mark function. One such construction is obtained by measuring the density of the [[Farey sequence|Farey numbers]] on the real number line. The question mark measure is the prototypical example of what are sometimes referred to as [[multifractal|multi-fractal measure]]s.
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| The question mark function maps rational numbers to [[dyadic rational|dyadic rational number]]s, meaning those whose [[Binary numeral system|base two]] representation terminates, as may be proven by induction from the recursive construction outlined above. It maps [[quadratic irrational]]s to non-dyadic rational numbers. It is an [[odd function]], and satisfies the functional equation {{math|1=?(''x'' + 1) = ?(''x'') + 1}}; consequently {{math|''x'' → ?(''x'') − ''x''}} is an odd [[periodic function]] with period one. If {{math|?(''x'')}} is irrational, then {{mvar|x}} is either [[algebraic number|algebraic]] of degree greater than two, or [[transcendental number|transcendental]].
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| The question mark function has [[Fixed point (mathematics)|fixed point]]s at 0, 1/2 and 1, and at least two more, symmetric about the mid-point. One is approximately 0.42037.<ref name=Fin442/>
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| The graph of Minkowski question mark function is a special case of fractal curves known as [[de Rham curve]]s.
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| ==Conway box function==
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| {{See also|Sawtooth wave}}
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| The ? is invertible, and the [[inverse function]] has also attracted the attention of various mathematicians, in particular [[John Horton Conway|John Conway]], who discovered it independently, and whose notation for {{math|?<sup>−1</sup>(''x'')}} is {{mvar|x}} with a box drawn around it: <span style="padding:0px 3px 1px; border:2px; border-color:black; border-style:solid; margin:1px">{{mvar|x}}</span> The box function can be computed as an encoding of the [[binary numeral system|base two]] expansion of <math>(x-\lfloor x \rfloor)/2</math>, where <math>\lfloor x \rfloor</math> denotes the [[floor function]]. To the right of the point, this will have {{math|size=120%|''n''<sub>1</sub>}} 0s, followed by {{math|size=120%|''n''<sub>2</sub>}} 1s, then {{math|size=120%|''n''<sub>3</sub>}} 0s and so on. For <math>n_0 = \lfloor x \rfloor</math>,
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| :{{bigmath|1=<span style="padding:0px 3px; border:2px; border-color:black; border-style:solid; margin:1px">''x''</span> = [''n''<sub>0</sub>; ''n''<sub>1</sub>, ''n''<sub>2</sub>, ''n''<sub>3</sub>, … ],}}
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| where the term on the right is a [[continued fraction]].
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| ==See also==
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| * [[Pompeiu derivative]]
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| ==Notes==
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| {{reflist}}
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| ==Historical references==
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| * H. Minkowski, ''Verhandlungen des III. internationalen Mathematiker-Kongresses in Heidelberg'', (1904) Berlin.
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| * A. Denjoy, ''Sur une fonction réelle de Minkowski'', J. Math. Pures Appl. '''17''' (1938) p. 105–151.
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| ==References==
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| *{{citation
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| | last = Alkauskas | first = Giedrius
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| | publisher = [[University of Nottingham]]
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| | series = Ph.D. thesis
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| | title = Integral transforms of the Minkowski question mark function
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| | url = http://etheses.nottingham.ac.uk/641/
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| | year = 2008}}.
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| *{{citation
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| | last1 = Bibiloni | first1 = L.
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| | last2 = Paradis | first2 = J.
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| | last3 = Viader | first3 = P.
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| | doi = 10.1006/jnth.1998.2294
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| | journal = Journal of Number Theory
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| | pages = 212–227
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| | title = A new light on Minkowski's ?(x) function
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| | issue = 2
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| | url = http://www.econ.upf.es/en/research/onepaper.php?id=226
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| | volume = 73
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| | year = 1998 | zbl=0928.11006 }}.
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| *{{citation
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| | last1 = Bibiloni | first1 = L.
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| | last2 = Paradis | first2 = J.
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| | last3 = Viader | first3 = P.
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| | journal = Journal of Mathematical Analysis and Applications
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| | pages = 107–125
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| | title = The derivative of Minkowski's singular function
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| | issue = 1
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| | url = http://www.econ.upf.es/en/research/onepaper.php?id=378
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| | volume = 253
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| | year = 2001
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| | doi = 10.1006/jmaa.2000.7064 | zbl=0995.26005 }}.
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| *{{citation
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| | last = Conley | first = R. M.
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| | publisher = [[West Virginia University]]
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| | series = Masters thesis
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| | title = A Survey of the Minkowski ?(x) Function
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| | year = 2003}}.
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| *{{citation
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| | last = Conway | first = J. H. | author-link = John Horton Conway
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| | contribution = Contorted fractions
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| | edition = 2nd
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| | location = Wellesley, MA
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| | pages = 82–86
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| | publisher = A K Peters
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| | title = On Numbers and Games
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| | year = 2000}}.
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| *{{citation | last=Finch | first=Steven R. | title=Mathematical constants | series=Encyclopedia of Mathematics and Its Applications | volume=94 | location=[[Cambridge]] | publisher=[[Cambridge University Press]] | year=2003 | isbn=0-521-81805-2 | zbl=1054.00001 }}
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| *{{citation | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| ==External links==
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| * [http://uosis.mif.vu.lt/~alkauskas/minkowski.htm An extensive bibliography list]
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| *{{citation
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| | last = Vepstas | first = L.
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| | title = The Minkowski Question Mark and the Modular Group SL(2,Z)
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| | url = http://www.linas.org/math/chap-minkowski.pdf
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| | year = 2004}}
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| *{{mathworld|urlname=MinkowskisQuestionMarkFunction|title=Minkowski's Question Mark Function}}
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| * [https://gist.github.com/pallas/5565556 Simple IEEE 754 implementation in C++]
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| [[Category:Fractal curves]]
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| [[Category:Continued fractions]]
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| [[Category:Special functions]]
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| [[Category:Continuous mappings]]
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| [[Category:Articles with example C code]]
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