Rotation number elementary theory for Birkhoff curves has been constructed. Geometrical (dynamical) and numerical properties for Birkhoff curves being more than two regions common boundary has been studied. Topological number invariants with respect to a dissipative dynamic system on the plane possessing the Birkhoff curve property have been discussed. Simple allocation algorithm of natural numbers has been applied, so that its Schnirelmann density is equal to the rotation number for a region. If the region boundary is a Birkhoff curve then the sequence contains an additive basis zero Schnirelmann density. The basis contains an arbitrary long arithmetic progression. Rotation numbers for regions are defined to be different additive bases zero Schnirelmann density.