Timeline for How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Jan 9, 2017 at 18:52 | comment | added | Will Sawin | There is a lower bound by norms: There are at most $n$ conjugates of this sum, each of size at most $\log n$, and the product of the conjugates is a nonzero integer, hence at least $1$, so the number is at least $1/(\log n)^n = e^(-n \log \log n)$. Of course this is quite far from the upper bound and conjectural lower bound that Vesselin Dimitrov gave. | |
| Jan 9, 2017 at 18:27 | comment | added | Vesselin Dimitrov | A pigeonholing argument shows it can be as small as exponential in $-(\log{n})^2$. It should be bounded below at least by $e^{-o(n)}$ as $n \to \infty$, but already this is an unsolved problem, even for sums of five $n$-th roots of unity. Non-trivial upper bound can be proved for the number of extremely small sums. | |
| Jan 9, 2017 at 17:32 | history | asked | kodlu | CC BY-SA 3.0 |