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

[franktaw at netscape.net]

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


-----Original Message-----
From: 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 
 
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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