Timeline for Lower bound for some sums of roots of unity
Current License: CC BY-SA 4.0
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| Nov 22, 2018 at 21:18 | comment | added | shurtados | @GerryMyerson: Thanks for the observation. I'm more interested in the random case, when the size of $A, B$ is comparable to $n$, and even for more general choices of $x_i$, but always keeping $x_i$ very small compared to $n$. | |
| Nov 22, 2018 at 15:01 | comment | added | shurtados | @Seva: Yes, that's what I want. I modified the statement to make it hopefully more clear. | |
| Nov 22, 2018 at 14:58 | history | edited | shurtados | CC BY-SA 4.0 |
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| Nov 22, 2018 at 11:57 | comment | added | Gerry Myerson | Write $S_n=A-B$, where $A$ collects all the $+1$ terms, $B$, all the $-1$ terms. Then $S+2B=A+B=0$, so $B=-(1/2)S$, so $S$ is tiny if and only if $B$ is tiny. So I don't think "taking all the roots of unity" makes the problem any easier than taking any old sum of $n$th roots of unity. | |
| Nov 22, 2018 at 10:10 | comment | added | Seva | What do you mean writing "given that there are $2^n$ sums, I am expecting a better lower bound"? Don't you need a uniform lower bound for all these sums? That is, do you want to show that none of the $2^n$ sums are small in absolute value? | |
| Nov 22, 2018 at 6:36 | history | edited | shurtados |
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| Nov 22, 2018 at 0:28 | comment | added | shurtados | Thank you, I saw that while writing the question, so it seems the question is very difficult, my hope is that because I'm taking all the roots of unity something different can be said, for example when $x_i$ are characters, the absolute value of the sum is $\sqrt(p)$ (If I'm correct). | |
| Nov 22, 2018 at 0:16 | comment | added | Mark Lewko | See: mathoverflow.net/questions/46068/… for a related discussion. | |
| Nov 22, 2018 at 0:08 | history | asked | shurtados | CC BY-SA 4.0 |