# Does coarse coding with radial basis function generate fewer features?

I am learning about discretization of state space when applying reinforcement learning to continuous state space. In this video the instructor at 2:02 the instructor says that one benefit of this approach (radial basis functions over tile coding) is that "it drastically reduces the number of features". I am not able to deduce this in the case of a simple 2D continuous state space.

Suppose we are dealing with 2D continuous state space, so any state is a pair $$(x,y)$$ in the Cartesian space. If we use Tile Coding and select $$n$$ tiles, the resulting encoding will have $$2n$$ features, consisting of $$n$$ discrete valued pairs $$(u_1, v_1) \dots (u_n, v_n)$$ representing the approximate position of $$(x,y)$$ in the frames of the $$n$$ 2-D tiles. If instead we use $$m$$ 2-D circles and encode using the distance of $$(x,y)$$ from the center of each circle, we have $$m$$ (continuous) features.

Is there a reason to assume that $$m < 2n$$?

Furthermore, the $$m$$-dimensional feature vector will again need discretization, so it is unclear to me how this approach uses fewer features.