Abstract: We explore the phase diagram of approximation rates for deep neural networks and prove several new theoretical results. In particular, we generalize the existing result on the existence of deep discontinuous phase in ReLU networks to functional classes of arbitrary positive smoothness, and identify the boundary between the feasible and infeasible rates. Moreover, we show that all networks with a piecewise polynomial activation function have the same phase diagram. Next, we demonstrate that standard fully-connected architectures with a fixed width independent of smoothness can adapt to smoothness and achieve almost optimal rates. Finally, we consider deep networks with periodic activations ("deep Fourier expansion") and prove that they have very fast, nearly exponential approximation rates, thanks to the emerging capability of the network to implement efficient lookup operations.