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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".

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I'm not really sure what this question wants. From a quick read of the paper (available here: the answer is "yes, but I don't see why we need $\|f\|_{L^\infty}\leq 1$". – Matthew Daws Dec 13 '10 at 15:49
The question wants the definition of uniformly almost periodic functions in the paper and not in the infinitary ergodic theory setting. – user4949 Dec 13 '10 at 16:38
I think uniform almost periodicity here mean being an element of $UAP^d$. – Dea Taha Dec 14 '10 at 11:44

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.

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+1, as I agree (and I said this in my comment above as well!) – Matthew Daws Dec 28 '10 at 14:50

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

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Well, the setting in Tao's proof is finitary, in the sense that the function $f$ above is defined on a finite cyclic group $Z_N$. Thus, continuous functions on the Bohr compactification of the reals might not be the answer. – user4949 Dec 13 '10 at 14:54

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