Timeline for Large deviations for trigonometric polynomials
Current License: CC BY-SA 4.0
5 events
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| Oct 14, 2019 at 15:16 | comment | added | mathworker21 | It should be noted that one also wants the $a_l$'s large so that the second factor is indeed typically of size $\sqrt{L}$ (obviously for $x$ close to $0$, the second factor is of size $L$ -- the point is the "close to" depends on the size of the $a_l$'s). | |
| Oct 11, 2019 at 16:29 | comment | added | Terry Tao | If one shifts all the $n_i$ in my example by a large factor $R$ then they will all lie in $[R,2R]$ but the distribution of the magnitude of the trignometric sum is unchanged. So if $R$ is allowed to be really large (E.g. exponentially large in $N$) this doesn't help at all. However one might be able to do something in the opposite regime when $R$ is comparable to $N$ and there is not enough "room" to have independent behaviour. | |
| Oct 11, 2019 at 7:52 | comment | added | Kurisuto Asutora | Thank you, this is very helpful. If I understand correctly, then in probabilistic terms the polynomial here is constructed as a sum of (essentially) independent objects, with the independence coming from the separation of frequencies. Do you have any intuition if a similar example could also be constructed without resorting to this sort of independence? More specifically, is the $L^2$ bound still sharp if we additionally could assume that the $n_1, \dots, n_N$ are of comparable size, such as $n_1, \dots, n_N \in [R,2R]$ for some $R$? | |
| Oct 11, 2019 at 7:43 | vote | accept | Kurisuto Asutora | ||
| Oct 11, 2019 at 2:42 | history | answered | Terry Tao | CC BY-SA 4.0 |