(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.