Global maximum of ?(x)-x

[Joseph Biberstine]

	Indeed.  I have been using the following Mathematica function for rough 
work:

mqf[x_] := If[x < 0, -mqf[-x], (cf = ContinuedFraction[(*some bug causes 
certain decimals to be misinterpreted*)Rationalize[x]]; cf[[1]] + 
Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}])];

	(It is interesting to note there is something wrong with Mma 5.1 or my 
installation specifically that, without the Rationalize comment noted in 
the comment above, even rational numbers can be completely bugged.  For 
example, without the Rationalize command above, I get mqf[0.4]=1/2 which 
is wrong.  mqf[2/5] yields the correct value of 3/8.  Thus why I say 
"rough" work above.)	
	This implementation is intended for rational x only (mqf is for mq 
finite cf).  Generalizing to a function approximating reals with 
non-terminating c.f.'s is trivial.  Finally, recall that Mma one-indexes 
its lists.
	My implementation similarly yields the correct special values and 
graphs appropriately.  I was going to attach a pair of images to my last 
message but worried the list might bounce for anti-solicitation reasons, 
but I think I'll try this time.
	I'm sure that the function is fractal (beautifully I might add).  This 
is evident from the plot.

	I'm glad you also noticed how well dyadics approximate many interesting 
constants under application of Minkowski's terrific function.  This of 
course stems from that large terms in an input's c.f. expansion (such as 
292 early in Pi and so 1/Pi) results in a drastic reduction of all 
following summands.  For further reference, see my OEIS submissions 
under 
http://www.research.att.com/~njas/sequences/?q=expansion+minkowski+biberstine&sort=2 
, especially http://www.research.att.com/~njas/sequences/A119929 ! 
Also, a notice to all: I recently corrected/modified several of these 
entries and Neil hasn't had time to apply the changes yet, so I suggest 
these be taken with a grain of salt for a few weeks (specifically,
A119928 and A119929 which I warn you are entirely incorrect due to a 
code artifact (namely I had been using a conditional on 
Element[x,Rationals] in my function for unification purposes and, of 
course, Khinchin does not (yet) have a Boolean evaluation in Mma).

Gerald McGarvey wrote:
 >
 > I defined the following function in PARI/GP for Minkowski's question
 > mark function,
 > using formula (1) on
 > http://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html
 > then tested it using the special values shown on the web page, it checks
 > out ok for
 > the special values.
 >
 > 
mq(x)=cf=contfrac(x);return(sum(k=1,length(cf)-1,(-1.)^(k-1)/2^(sum(m=2,k+1,cf[m])-1))) 

 >
 >
 > contfrac(x) creates an array, the rest uses the array.
 > Something similar could be done in Mathematica or Maple.
 >
 > then defined f(x) = mq(x)-x for ?(x)-x
 >
 > I used a crude approach (using print in a for loop) to find larger
 > values for
 > ?(x)-x near the value .79285714 and found (using a precision of \p 115)
 >
 > f(.79289)    = .142588184488601982593536376953125 (not sure if exact)
 >
 > f(.79285714) = .1425676646875 (not sure if exact)
 >
 > so I think f(.79285714) is near a local maximum, but that the function
 > then decreases before going to the global maximum, wherever that is.
 > The problem is that f(x) is not strictly increasing.
 > Could it be fractal in nature? If so, finding the global maximum could
 > be very difficult, but maybe there is an easy way. It could require
 > a bit of programming that iteratively creates arrays of values and finds
 > the maximum of the array to narrow the search. How fine-grained would
 > each step need to be, how erratic is ?(x)-x ?
 >
 > Using the mq function I get the following puzzling values.
 > Are these values accurate or due to a calculation error?
 >
 > ?(1/Pi) =
 > 
0.248046904802322387695312500000000000000000000000000000000000000000000000000000000000000000000002403016875... 

 >
 >
 > ?(1/Pi^3) =
 > 
0.000000000931322574615478515624999999999999999999999999998884213294804113213548547849861136557254326556794... 

 >
 >
 > - Gerry
 >
 > At 07:10 PM 6/10/2006, Joseph Biberstine wrote:
 >>         Not clever enough to figure this out a reliable computation of
 >> this myself.  Please help find enough accurate terms for OEIS and I
 >> will of course credit you.
 >>
 >>         Consider f(x) := ?(x)-x where ?(x) is Minkowski's question
 >> mark function.
 >>         Recently I've wondered at what value xmax f will obtain its
 >> global maximum (mod 1).  Specifically I'm curious how the c.f.
 >> expansion for xmax grows (based on the definition of f it should make
 >> for a very interesting sequence!)
 >>         To the few places I've been able to calculate, neither c.f.
 >> nor decimal expansions for xmax or f(xmax) are in OEIS.  Note the c.f.
 >> expansion looks about as we'd expect so far (that is, when
 >> zero-indexed, odd entries are 1 and even entries slowly grow).
 >>
 >> xmax =~ {0; 1, 3, 1, 4, 1, 4, ?1, ...} =~ 0.79285714...
 >>
 >> - Joseph
 >

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