Global maximum of ?(x)-x (consolid'd replies)

[Joseph Biberstine]

**********
franktaw at netscape.net wrote:
 > <snip> I propose calling this the Pierce function. <snip>

Here's the applicable Mathematica code:

(* I like to do these first *)
Clear[pierceapx];
Off[ContinuedFraction::incomp]; (*kills msg for cf expansions 
terminating before arbitrary precision*)
(* Here's the meat *)
pierceapx[x_] := (cf = ContinuedFraction[x,(*ARBITRARY*)100]; 
Sum[(-1)^(k)/Product[Sum[cf[[i]], {i, 1, j}], {j, 2, k}], {k, 2, 
Length[cf]}]);

It yields the values per your indication.  Attached are a graph of the 
Pierce function less the identity on 0<x<1 and the start of the symbolic 
formula for Pierce on the continued fraction [a0; a1, a2, a3, ...].

I agree Frank; this is a terrific function.



**********
franktaw at netscape.net wrote:
 > Here's a little more info.
 >
 > The solution Pierce(x) = x near 0.355 is approximately 0.35446930411246,
 > with continued fraction  2,1,4,1,1,2,3,1,2,1,1,1,... (and hence Pierce
 > expansion 2,3,7,8,9,11,14,15,17,18,19,20,...).  The global maximum
 > appears to be near 0.57187500456342, with continued fraction
 > 1,1,2,1,44,1,2138,... (I'm not 100% sure that that isn't just a local
 > maximum, and the 2138 might be off.  The value at that point is about
 > 0.60050000466366.
 >
 > Franklin T. Adams-Watters

I do hope you plan to submit these to OEIS soon!



**********
Gene Smith wrote:
 > I think it would be of considerable interest to push this farther.  Has
 > anyone considered using box rather than the ? function? I continue to
 > suspect it would be easier.

Folks keep suggesting Conway's box, but I'm afraid I'm not myself 
schooled enough to see the connection between extrema of f(x)-x and 
x-f_inv(x).  Since I cannot do this one myself, I am again submitting 
the (inelegant) Mathematica code for Conway's box.  Here's an open req 
hoping someone will take up this tack.

(* exact; for dyadic x *)
cbd[x_] := (y = Mod[x,1]; If[y == 0, x, d = RealDigits[y, 2]; nd = 
Join[Table[0, {i, 0, -d[[2]]}], d[[1]]]; 
FromContinuedFraction[Join[{Floor[x]}, Length /@ Split[nd]]]]);

(* approx (change magic constant where marked as desired); for all x *)
cbn[x_] := (y = Mod[x, 1]; If[y == 0, x, d = RealDigits[y, 2,(*arb 
precn*)200]; nd = Join[Table[0, {i, 0, -d[[2]]}], d[[1]]]; 
FromContinuedFraction[Join[{Floor[x]}, Length /@ Split[nd]]]]);

- Joseph Biberstine

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