Thanks, Hans. In fact I saw an argument in Wikipedia (< http://en.wikipedia.org/wiki/Khinchin's_constant >), which is claimed to be much simpler than Khinchin's original reasoning. ((( But it omits explaining the "hard part" -- which is proving that when restricted to the irrational numbers J = (0,1) - Q, the transformation T: J -> J defined via CF convergents as: T([0; a_1, a_2, . . .]) = [0; a_2, a_3, . . .] has no invariant set of intermediate measure* (i.e., not 0 or 1). Of course, T(x) = frac(1/x). Gauss discovered that this function has an invariant measure given by mu(A) = (1/ln(2))*Integral_{A} of 1/(1+x) dx. which is kind of fun to sit down and prove for oneself. ))) --Dan ___________________________________________________________________ *Whether this invariant set's measure is taken as Lebesgue or Gauss's mu makes no difference to its being "of intermediate measure". On 2012-12-24, at 10:59 AM, Hans Havermann wrote:
Dan Asimov:
It's amazing that Khinchin found that for almost all positive reals x, the GM of the convergents of its CF expansion are independent of x.
Is there a simple reason why this should be true?
It's not simple to me but here is Khinchin's translated 1935 argument:
http://chesswanks.com/txt/Khinchin.pdf
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