Timeline for How often does the omega theorem hold?
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| Apr 7, 2022 at 5:19 | comment | added | GH from MO | @WSao I recommend that you study Bohr's book "Almost periodic functions". Consider $f(x):=\sum_{n=1}^N\max\left(0,1-q\left\|c_n x\right\|\right)$, where $c_n:=\gamma_n/(2\pi)$. This is a sum of continuous periodic functions, hence it is almost periodic (as shown by Bohr). In particular, every sufficiently long interval contains $U$ such that $\sup_x|f(x+U)-f(x)|<1$. Take such a $U$. From $f(0)=N$ it follows that $f(U)>N-1$, so none of the $N$ terms in the definition of $f(x)$ vanishes at $U$. That is, $\|c_n U\|<1/q$ holds for $n\in\{1,\dotsc,N\}$. This is the claim that Ingham uses. | |
| Apr 5, 2022 at 2:04 | comment | added | W Sao | Thanks for the references. Ingram's paper make reference to a theorem of Bohr on diophantine approximation. I'm new to this subject; could you give me an English reference of Bohr's result, please? Thanks! | |
| Mar 23, 2022 at 20:30 | vote | accept | W Sao | ||
| Mar 23, 2022 at 1:32 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Mar 23, 2022 at 0:56 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Mar 23, 2022 at 0:38 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Mar 22, 2022 at 23:32 | history | answered | GH from MO | CC BY-SA 4.0 |