Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

(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.

share|improve this question
Was it related to simple continued fractions? –  Douglas Zare Apr 6 '11 at 5:57
add comment

1 Answer

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.

share|improve this answer
+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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.