Existence of Large Independent-Like Sets

Let $G$ be a compact abelian group and $\Gamma$ be its discrete dual group. For $N \in \mathbb{N}$, we define a class of independent-like sets, $N$-PR sets, as a set 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 structures and prove the existence of large $N$-PR sets.