The ergodic theorems can be seen as improvements of the Poincaré recurrence theorem, because they relate the time average with the space average. However, the Poincaré theorem is treated first because it is not hard to prove and it is very nice.

I got the above PDF version from Shigeki Akiyama and I am very glad that putting it on my website is useful to some people. It would be great if more people did this because trying to find a paper that is neither on the internet or in libraries can be extremely frustrating!

Note that this solution yields an answer with $2k + 11$ pieces, for all $k \geq 0$.