Global maximum of ?(x)-x: a related problem

[franktaw at netscape.net]

You folks might try also looking at another, similar function I 
discovered recently.  This relates to the very nice Pierce expansion.  
Every irrational number between 0 and 1 has a unique representation as 
an alternating sum:

x = 1/a_1 - 1/(a_1 a_2) + 1/(a_1 a_2 a_3) - ...,

where the sequence of a_i's are strictly increasing.  Rational numbers 
in this range have two representations, one ending in a-1, a and the 
other with just a.

So take a number between zero and one, get its continued fraction, take 
the cumulative sums, and regard that as a Pierce expansion.  Voila - an 
increasing, one-to-one function from [0,1] to [0,1].  I propose calling 
this the Pierce function.

Some particular values:

phi^{-1} -> 1-e^{-1}
sqrt(2)-1 -> 1-e^{-1/2}

1/n and 1-1/n are fixed points, but there are others,  including one at 
about 0.355 (and, of course, another at about 0.645).  The global 
maximum for Pierce(x)-x appears to be near 0.428.

Franklin T. Adams-Watters

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