Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:43:40Z http://mathoverflow.net/feeds/question/49263 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49263/almost-periodic-functions-in-taos-ergodic-proof-of-szemeredis-theorem Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem unknown (google) 2010-12-13T14:22:07Z 2010-12-28T11:24:45Z <p>Do we say that a function $f$ is <em>uniformly almost periodic</em> 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 $f\in UAP^d$?</p> <p><em>Edit</em>: the proof in question appears in Tao's paper "A quantitative ergodic theory proof of Szemerédi's theorem".</p> http://mathoverflow.net/questions/49263/almost-periodic-functions-in-taos-ergodic-proof-of-szemeredis-theorem/49265#49265 Answer by follower for Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem follower 2010-12-13T14:38:21Z 2010-12-13T14:38:21Z <p>I think it means the continuous functions on the Bohr compactification of the reals.</p> <p><a href="http://en.wikipedia.org/wiki/Almost_periodic_function#Uniform_or_Bohr_or_Bochner_almost_periodic_functions" rel="nofollow">http://en.wikipedia.org/wiki/Almost_periodic_function#Uniform_or_Bohr_or_Bochner_almost_periodic_functions</a></p> http://mathoverflow.net/questions/49263/almost-periodic-functions-in-taos-ergodic-proof-of-szemeredis-theorem/50495#50495 Answer by Dee for Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem Dee 2010-12-27T14:20:20Z 2010-12-28T11:24:45Z <p>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.</p>