Global maximum of ?(x)-x

[Gerald McGarvey]

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