Kernelization and approximation of distance-$r$ independent sets on nowhere dense graphs

For a positive integer $r$, a distance-$r$ independent set in an undirected graph $G$ is a set $I\subseteq V(G)$ of vertices pairwise at distance greater than $r$, while a distance-$r$ dominating set is a set $D\subseteq V(G)$ such that every vertex of the graph is within distance at most $r$ from a vertex from $D$. We study the duality between the maximum size of a distance-$2r$ independent set and the minimum size of a distance-$r$ dominating set in nowhere dense graph classes, as well as the kernelization complexity of the distance-$r$ independent set problem on these graph classes. Specifically, we prove that the distance-$r$ independent set problem admits an almost linear kernel on every nowhere dense graph class.