Let $x\_1,\ldots,x\_n$ be $n$ complex numbers, and define $x\_I:=\sum\_{i\in I}x\_i$ for any set $I\subseteq[n]$. Finally, declare the family $(x\_1,\ldots,x\_n)$ to be "sumset-distinct" if the $2^n$ numbers $(x\_I)\_{I\subseteq [n]}$ are pairwise distinct. My questions are:


1. Has this notion been studied and if yes under what name?
2. Is there a simple characterization of sumset-distinct families?
3. Are there simple methods to determine whether a given family is sumset-distinct. For example, are the $n^{th}$ roots of unity $\left(e^{\frac{2ik\pi}{n}}\right)\_{1\le k \le n}$ sumset-distinct?



Sumset-distinct numbers
The set $\{x\_i\}$ is called in the literature a sum-distinct set.


There is a big open problem in the area, due to Erdős and Moser: determining $f(n)=\min\{ \max(S): |S|=n, S\subseteq \mathbb{N}, S \text{ sum-distinct}\}$. The current lower bounds on $f$ are of the form $f(n)\ge C 2^n/\sqrt{n}$. A trivial upper bound is $f(n)\le 2^{n-1}$ by taking $x\_i=2^{i-1}$.


There are some useful characterizations which were used in the literature to attack this problem. One characterization used by [Dubroff--Fox--Xu](https://arxiv.org/abs/2006.12988) is that if $\varepsilon\_i$ are i.i.d random variables taking the values $\pm 1$ with equal probabilities then $\sum \varepsilon\_i x\_i$ takes each of its $2^n$ possible values with probability $2^{-n}$ iff $S$ is a sum-distinct set.
[A related (earlier) observation of Elkies](https://www.sciencedirect.com/science/article/pii/0097316586901160) is that if $x\_i$ are integers valued then the Laurent polynomial $A(z)=\prod\_i(1+(z^{x\_i}+z^{-x\_i})/2)$ has a very particular coefficient structure if $S$ is sum-distinct (and in particular the constant coefficient is $1$).


