Existence of large independent-like sets
Let $G$ be a compact abelian group and $\varGamma $ be its discrete dual group. For $N \in \mathbb N $, we define a class of independent-like sets, $N$-PR sets, to be sets in $\Gamma $ such that every $\mathbb Z _N$-valued function defined on the set can be interpolated by a character in $G$.
These sets are examples of $\varepsilon $-Kronecker sets and Sidon sets. In this paper we study various properties of $N$-PR sets. We give a characterization of $N$-PR sets, describe their structure and prove the existence of large $N$-PR sets.