I apologise if this is well-known or straightforward.


Define the Fourier transform of the characteristic function of a subset $E\subseteq\mathbb{Z}\_n$ by
$$
\widehat{1\_E}(k)=\sum\_{a \in E} \exp(-2 \pi i ak/n).
$$
If $n$ is an odd prime, this sum is always nonzero, for all nonempty proper subsets $E$. Can one characterize sets $E$ for which this sum is nonzero for $n$ composite?



For which sets $E\subset \mathbb{Z}_n$ is $\widehat{1(E)}$ nonzero everywhere?
This is true if and only if the vector space generated by the translates of $E$ has dimension $n$, which is a purely physical space characterization. This is the discrete Fourier version of Wiener's tauberian theorem, and follows from the fact that the Fourier transform takes convolutions to products.


