Number of solutions for this diophantine equation using character theory?

Question

Let $G$ an abelian group with $\vert G\vert<+\infty$ and $N_1, ..., N_k$ , $k$ subsets of $G$. Let $a \in G$.

We want to find $\vert \{(n_1,...,n_k)\in N_1\times...\times N_k \ / \ n_1+...+n_k=a \}\vert$.

Apparently we can use the character theory to find this number. So if someone has references on that fact, it would be great.

Thanks in advance !

Answer 1

I guess what you mean is that the number of $k$-tuples in question can be written as $$ \frac1{|G|}\,\sum_{n_1\in N_1,\dotsc,n_k\in N_k} \sum_{\chi\in\widehat G} \chi(n_1+\dotsb+n_k-a) = \frac1{|G|} \sum_{\chi\in\widehat G} \chi(-a)\prod_{j=1}^k\widehat{N_j}(\chi), $$ where $$ \widehat{N_j}(\chi) = \sum_{n_j\in N_j}\chi(n_j) $$ are the (non-normalized) Fourier coefficients of the indicator function of $N_j$. The main term obtained when $\chi$ is the principal character is $$ \frac1{|G|}|N_1|\dotsb|N_k|, $$ to estimate the reminder term one needs some information about the sets $N_j$.