  In this paper, we develop achievability schemes for symmetric $K$-user
interference channels with a rate-limited feedback from each receiver to the
corresponding transmitter. We study this problem under two different channel
models: the linear deterministic model, and the Gaussian model. For the
deterministic model, the proposed scheme achieves a symmetric rate that is the
minimum of the symmetric capacity with infinite feedback, and the sum of the
symmetric capacity without feedback and the symmetric amount of feedback. For
the Gaussian interference channel, we use lattice codes to propose a
transmission strategy that incorporates the techniques of Han-Kobayashi message
splitting, interference decoding, and decode and forward. This strategy
achieves a symmetric rate which is within a constant number of bits to the
minimum of the symmetric capacity with infinite feedback, and the sum of the
symmetric capacity without feedback and the amount of symmetric feedback. This
constant is obtained as a function of the number of users, $K$. The symmetric
achievable rate is used to characterize the achievable generalized degrees of
freedom which exhibits a gradual increase from no feedback to perfect feedback
in the presence of feedback links with limited capacity.
