Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem - MathOverflow

Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem

2010-12-13T14:22:07Z

Do we say that a function $f$ is uniformly almost periodic in the aforementioned proof if $f$ is bounded (in the sense that $||f||_{L^\infty}\leq 1$) and that there exists a natural number $d>0$ such that $f\in UAP^d$?

Edit: the proof in question appears in Tao's paper "A quantitative ergodic theory proof of Szemerédi's theorem".

Answer by follower for Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem

2010-12-13T14:38:21Z

I think it means the continuous functions on the Bohr compactification of the reals.

http://en.wikipedia.org/wiki/Almost_periodic_function#Uniform_or_Bohr_or_Bochner_almost_periodic_functions

Answer by Dee for Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem

2010-12-27T14:20:20Z

As I mentioned in my comment above, a uniform almost periodic function (of order $d$) is an element of $UAP^d$ for some $d$. The bound $||f||_{L^\infty}\leq1$ is unnecessary.