  We consider density estimation for Besov spaces when each sample is quantized
to only a limited number of bits. We provide a noninteractive adaptive
estimator that exploits the sparsity of wavelet bases, along with a
simulate-and-infer technique from parametric estimation under communication
constraints. We show that our estimator is nearly rate-optimal by deriving
minimax lower bounds that hold even when interactive protocols are allowed.
Interestingly, while our wavelet-based estimator is almost rate-optimal for
Sobolev spaces as well, it is unclear whether the standard Fourier basis, which
arise naturally for those spaces, can be used to achieve the same performance.
