Question

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 1

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.

Answer 2

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