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(My apologies in advance; this is very open-ended but I ask leave to post regardless.)

I'm trying to recall a theorem on the fractional part of... some fairly natural class of sequences. It showed that the expected value is not 1/2, as might be assumed, but rather some smaller value (perhaps around 0.4). Unfortunately I can't think of what sorts of sequences these were, and that makes it quite hard to recall the theorem itself.

It was not about some contrived sequence like the Pisot/PV numbers. If I can think of additional details I will edit them in or add them as a comment.

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Was it related to simple continued fractions? – Douglas Zare Apr 6 '11 at 5:57
up vote 9 down vote accepted

A few possibilities come to mind including Benford's law.

Here is another: If $X$ is uniform on $[0,1]$ then $\frac {1}{X} - \lfloor \frac{1}{X} \rfloor$ is not uniform on $[0,1]$. For example, the density at $1/2$ is $\frac{4}{9} + \frac{4}{25} + \frac{4}{49} + ... = \frac{\pi^2}{2} - 4 \approx 0.9348.$ The expected fractional part of $1/X$ is $1-\gamma \approx 0.422784.$

See also the Gauss-Kuzmin-Wirsing operator.

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+1. You are good at mind reading... – Did Apr 6 '11 at 6:36
Amazing. The second result was the one I was thinking of. As Didier Piau says, you are good at mind reading! – Charles Apr 6 '11 at 12:50
I'm tempted to add a mind-reading tag to this problem, and suggest a Mindreader badge be instituted. – Gerry Myerson Jul 8 '11 at 1:28

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