Are there $x_n$'s and $y_n$'s such that $\sum_{n=1}^Ne^{-i2m\pi x_n}=\sum_{n=1}^Ne^{-i2m\pi y_n}$ for $1\geq x_n>y_n\geq0$

Asked Feb 10 '21 at 19:49

Active Feb 10 '21 at 20:16

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I am looking for the unique values of $x_n$ and $y_n$ ($n=1,\dots,N$) where $1\geq x_n>y_n\geq0$ such that the folloiwng holds $$ \sum_{n=1}^Ne^{-i2m\pi x_n}=\sum_{n=1}^Ne^{-i2m\pi y_n} $$ for all non-zero integers $0<m\leq N-1$. I think there is a connection to the property of the roots of unity (their sum equals $0$) but I am not sure how to utilize it.

Edit:

Another requirement is that the values of $x_n-y_n$ should be unique for all $n$.

edited Feb 10 '21 at 20:14

asked Feb 10 '21 at 19:49

nOp

The question is weird posed: if you take $x_{n}=y_{n}=1$ then you are done. – Davide Trono Feb 10 '21 at 20:05
1

It can not be the case though. I am looking for unique values. Also, $x_n>y_n$. – nOp Feb 10 '21 at 20:07
1

Adding conditions that totally change the question after someone has posted doesn’t invite one to answer your questions. – robjohn Feb 10 '21 at 20:20
The case where $x_n-y_n$ is the same for all $n$ leads to duplicate of the post I linked to in the original post. I assumed one would not consider that case for this question. – nOp Feb 10 '21 at 20:50

Are there $x_n$'s and $y_n$'s such that $\sum_{n=1}^Ne^{-i2m\pi x_n}=\sum_{n=1}^Ne^{-i2m\pi y_n}$ for $1\geq x_n>y_n\geq0$

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