Talk:Minkowski's question-mark function

About the derivatives

It does not have a well-defined derivative, in the classical sense, on the irrationals;

That's impossible! Any monotonic function has a derivative almost everywhere. --a_dergachev (talk) 03:50, 8 May 2009 (UTC)

You are absolutely correct. (See Monotonic function for example. I am not surewhat the intended statement is but I am deleting the formulation as it is now since it is clearly wrong. I am also deleting another confusing statement

The derivative vanishes on the rational numbers; however, since the rationals are a set of measure zero,

this vanishing of the derivative at the rationals is not in contradiction with the non-absolute continuity of the function.

There are non-absolutely continuous functions with derivatives that vanish almost everywhere (the Cantor function for example) making this statement incomprehensible.(Sorry about the confusing mark-up of my post.) --MathHisSci (talk) 12:43, 18 March 2010 (UTC)

Comments 2005

Just a comment, this has to be the worst example of bad choice of notation...what was he thinking?? Revolver 07:06, 21 May 2005 (UTC)

Dates back to the dawn of the discovery of pathological functions, at the turn of the 20th century. Maybe it was felt to be very confusing? And maybe he had a twisted sense of humour. linas 00:36, 22 May 2005 (UTC)

Why is ? any crazier than ! (asks one who shorthands sin and cos with $ and ¢)? Kwantus 2005 June 29 17:52 (UTC)

Worse yet, it conflicts with a much more useful notation, d?k for the binomial coefficient d!/(d−k)!k!, generalizing to ?⟨i₀,…,i_n⟩ for the multinomial coefficient of a multi-index. Without regard for degree we can write

$?I{\mathbf {x} }^{I}$

as the general term of a multinomial expansion, where the multi-index exponent, as usual, means

$x_{0}^{i_{0}}x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$,

and

$|I|=i_{0}+i_{1}+\cdots +i_{n}$

equals the degree. Ah well; there's little chance of confusion. --KSmrq^T 20:53, 2005 August 31 (UTC)

Question

The fractal and self-similar nature of the function is unclear. Exactly how does the modular group describe the self-similarity? AxelBoldt 03:52, 29 March 2006 (UTC)

Yes, its a sloppy statement; it is only the "period-doubling monoid" inside SL(2,Z) (and not the modular group PSL(2,Z)) that applies to period-doubling fractals. (Different people seem to call this monoid different names). Consider the operators R and S:

$[S?](x)=?(x/1+x)=?(x)/2$

Note that the above is a self-symmetry for x in [0,1]: the question mark on the interval [0,1/2] is a half-size of the whole thing. Let R be a reflection:

$[R?](x)=?(1-x)=1-?(x)$

Then R and S generate the monoid: that is, any string of the form

$S^{m}RS^{n}RS^{p}...$

for positive integers m,n,p ... is a self-symmetry of the question mark. The requirement that m,n,p be poistive is what makes it a monoid,nt a group. linas 02:31, 30 March 2006 (UTC)

I fixed the article. The relationship to SL(2,Z)/PSL(2,Z) etc. is not hard but has some subtle confusions.linas 03:12, 30 March 2006 (UTC)

question

Is this function really absolutely continuous?

The Denjoy's article says it's purely singular. (See also mathworld) --a_dergachev (talk) 03:50, 8 May 2009 (UTC)

question/correction? =

When reading the page 84 of my copy of John Conway's "On Numbers and Games" (2nd edition, 2001, A K Peters, Ltd) I see at the top:

+---------------+
|               |
| (1+sqrt(5))/2 |  =  5/3
|               |
+---------------+

The function here called [x] is traditionally called "Minkowski's Question-Mark Function," and has interesting analytic properties.

(Where [x] is my ascii rendition of "x in a box"). So Conway _does not_ indicate the inverse of Minkowski's ? with a box as is claimed in the article, but the Minkowski's ?-mark function itself!

BTW, I created a new index entry to Sloane's OEIS for related sequences: http://www.research.att.com/~njas/sequences/Sindx_Me.html#MinkowskiQ You may add it to the external links section.

Yours, Antti Karttunen, his-firstname.his-surname@gmail.com

PS. You (Linas) erroneously claim on your "Wacky Thoughts" page that Minkowski's Question mark-function maps algebraic numbers to rationals, whereas it's only the "quadratic surd" -subset of them. (Like correctly explained on this page.)

iw

Is the interwiki link to Russian correct? --Meldor 00:10, 8 November 2007 (UTC)

Algorithm to compute ?(x)

Some time ago, I wrote a program to compute ?(x) and made a graph of it (much like the one in the article) before I even knew what the function was called. I just wanted to see what this function looked like whose derivative was zero at all the rationals. Since this function is continuous, defining it at the rationals is sufficient to define it for all the reals. An algorithm to compute the function (or more precisely, to approximate it to any desired degree of accuracy) would be nice in the article, and would also serve as an alternative definition. Something on the order of:

a := floor(x)

b := a + 1

d := 1

y := a

while (d > tolerance)

c := mediant(a,b)

d := d / 2

if (x >= c) then

y := y + d

a := c

else

b := c

output y

Of course a, b, and c have to be implemented as pairs of integers, and I like them to be proper fractions in practice. 198.145.196.71 (talk) 10:02, 2 March 2008 (UTC)

Fourier Transform

Since this function is periodic, I'm very curious what the fourier series of it would look like. Anyone know? Timeroot (talk) 19:33, 19 January 2010 (UTC)

Connection with SL(2,Z)

I think the connection given with $SL(2,Z)$ in the opening paragraph of "Recursive definition for rational arguments" is somewhat badly written, for reasons as below. If $p/q<r/s$ are two rational numbers the $rq-ps>0$ and so if the determinant is $\pm 1$, then in fact $rq-ps=1$.

If $\begin{pmatrix} r & p \\ s & q\end{pmatrix}$ is a matrix with positive integer entries such that $rq-ps=1$, then we can use the Euclid division algorithm to show that it can be uniquely written in the form $A^{a_1}B^{b_1}A^{a_2}B^{b_2}\dots A^{a_k}B^{b_k}$ where $a_i$ and $b_i$ are non-negative integers and all except possibly $a_1$ and $b_k$ are non-zero as well. This is what gives the connection of such matrices with continued fractions. — Preceding unsigned comment added by 210.212.36.65 (talk) 09:00, 30 March 2012 (UTC)

Mathematica implementation/hack

In case anyone looks for a short Mathematica one-liner implementing ?(x) for numerical values (I guess that would mean: for rationals) :

qm[x_] := Abs@Fold[#2 - #1 &, Floor@x, 2*2^-Accumulate@Rest@ContinuedFraction@x]

Summsumm2 (talk) 16:08, 4 February 2013 (UTC)

Conway box function and Sawtooth wave question

The Conway box function section says See also: Sawtooth wave, but neither mentions the other one. What is the connection? How reading about Sawtooth wave allows better or deeper understanding of a boxed question mark function? --CiaPan (talk) 10:22, 22 April 2015 (UTC)

I also did not see the link. I suggest this link be deleted. Also, reading the book of Conway it looks like his function is actually Minkowski's question mark, and not its inverse bijection (figure 8, page 84 there). Saros (talk) 22:26, 3 December 2017 (UTC)

A box

I could not find a single LaTeX command for a box to surround a given expression, however I discovered a quite simple way to make a box with an array:

\begin{array}{|c|} \hline EXPR \\ \hline \end{array}

Result:

${\begin{array}{|c|}\hline EXPR\\\hline \end{array}}$

The {|c|} part declares a single column in the array with centered contents and vertical lines on both sides. Commands \hline add horizontal lines above and below the (single) row. Command \\ terminates the row so that the bottom line can be drawn.
Happy editing, everybody! :) CiaPan (talk) 10:50, 22 April 2015 (UTC)

Minor language mistakes fixed, thanks to the notice by Steue at my talk page. --CiaPan (talk) 15:23, 19 July 2019 (UTC)

Salem's problem

There were two solutions posted on Salem's problem during Jan 2015. One was a special case of an earlier result by Jordan and Sahlsten (arXiv:1312.3619), that was submitted to arXiv on December 12, 2013 and accepted to the journal Mathematische Annalen on February, 2015, according to the arXiv version from February 2015. The earlier version of the paper by Jordan and Sahlsten did not mention Salem's problem and the application was pointed out by Persson (arXiv:1501.00876) in a note from January 5, 2015. Yakubovitch's answer (arXiv:1501.00141) was submitted to arXiv on December 31, 2014, and the corrected revision of that article was posted to arXiv on April 15, 2015. — Preceding unsigned comment added by 79.176.37.159 (talk) 06:45, 13 May 2015 (UTC)

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Conway box function

I have a copy On Numbers and Games, and the box function is clearly the Minkowski question mark function itself, not its inverse. I suggest removing the whole "Inverse" section from the article. Reyk _YO! 07:05, 11 September 2020 (UTC)

• Y- since there were no objections I removed it. Reyk _YO! 16:48, 10 October 2020 (UTC)

?(2/3)

Either the continued-fraction—binary definition is wrong, or I'm misunderstanding something.

The continued fraction representation of 2/3 is [0;1,2]. This run-length-encodes the binary number 0.011₂, which equals 3/8.

However, by the finite-sum equation given (and confirmed by WolframAlpha), ?(2/3) = 3/4 = 0.11₂. This would appear to correspond to either [0;0,2]=2 or [0;2]=1/2.

(Am I doing the whole process backwards? 2/3 = 0.1010101...₂ ⇒ [0;1,1,1,1,...] = 1/φ. That's not right either.)

What's going on?

— Sonata Green^(talk) 01:07, 13 April 2023 (UTC)

@Sonata Green There's an extra factor of 2 in front of the sum. The correct procedure is as follows:

1. Convert x to its continued fraction representation. Example: 2/3 = [0; 1, 2].
2. Interpret the numbers in the fractional part as sequences of alternating 0's and 1's, starting with 0 and ending with an infinite sequence of either zeros or ones: 011000... (1 zero and 2 ones).
3. Add a dot after the first zero and interpret the result as binary: 0.11₂ = 3/4.

Here's another example: x = 3/5 = [0; 1, 1, 2] = → 0100111... ⇒ ?(3/5) = 0.100111...₂ = 0.101₂ = 5/8.

Hugo Spinelli (talk) 10:10, 17 October 2023 (UTC)

Then why does [3;2,1,2,1,4,5,...] generate 11.001001000011111... rather than 11.01001000011111...? — Sonata Green^(talk) 21:34, 17 October 2023 (UTC)

@Sonata Green The example is wrong. It should be $[3;3,1,2,1,4,5,...]$. See WolframAlpha, for instance. One of the references given (Pytheas Fogg, 2002) appears to make the same mistake. The other one (Finch, 2003) is very explicit that the first sequence of zeros is $(a_{1}-1)$-long. —Hugo Spinelli (talk) 17:33, 20 October 2023 (UTC)

Where do the matrices come from in the "Periodic orbits as continued fractions" section?

I tried implementing this myself and ran into problems with the matrices given. Deriving them on my own, I got different matrices, and I updated the article with the ones I derived. I can't actually find where they appear in the cited reference, though. — Flipdisk (talk) 08:30, 27 March 2024 (UTC)

Actually, now that I read the earlier parts of the article more carefully, this section does seem to make sense in context, so I reverted my changes. But I still don't see where it appears in the source material. Flipdisk (talk) 08:32, 27 March 2024 (UTC)