Ideals of nowhere dense sets in some topologies on positive integers

We investigate the ideals of nowhere dense sets in three topologies on N (namely, the Furstenberg's, Golomb's, and Kirch's topology) related to arithmetic progressions. In particular, we explore relationships between these ideals, and show that each of them has a topological representation and cannot be extended to a summable ideal. Moreover, we study a related notion of Marczewski-Burstin countable representability.