Let be an ergodic measure-preserving system, and let be measurable. Suppose that

for some . Then there exists a set of measure at least such that .

With Corollary 1 as a special case, and a very similar proof.

]]>*[Huh, I thought I had fixed that already. In any case, fixed again – T.]*

*[Corrected, thanks – T.]*

I think only finitely many of the sets have nonzero measure; we may put in points with zero measure without changing anything.

Did you ever intend to put all your ergodic lectures somehow in one file, so that it’s easily printable??? I’d be VERY interested.

]]>*[Corrected to arctangent – T.]*