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

[franktaw at netscape.net]

Correction: 0.428 is the apparent global minimum of Pierce(x) - x; 
0.572 would then be the global maximum.

Franklin T. Adams-Watters


-----Original Message-----
From: franktaw at netscape.net
To: Gerald.McGarvey at comcast.net; jrbibers at indiana.edu
Cc: seqfan at ext.jussieu.fr
Sent: Wed, 14 Jun 2006 13:50:50 -0400
Subject: Re: Global maximum of ?(x)-x: a related problem

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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