What can one say about the set of continued fractions $[0;a\_1,a\_2,\ldots]$, where $a\_1,a\_2,\ldots$ are a *permutation* of the set of natural numbers?



Continued fractions using all natural integers
Yeah, it has measure 0 by what Qiaochu said, although I strongly suspect there's probably a more elementary way to prove this (or at least give a strong heuristic argument.)


It's also uncountable, which means that in particular: 1. There exist numbers of this type that are transcendental, 2. There exist numbers of this type that are uncomputable, 3. It's very very very very unlikely that you'll be able to come up with a nontrivially different characterization of them.


They don't form any algebraic structure of any significance as far as I can tell (not a field, not a ring, I'm 100% sure but not completely convinced not a group), they're obviously not dense in the reals...


In essence, these are pretty much a completely typical uncountable set of measure 0, as far as I can tell. 


