  The response of particle density to a dilation of a periodic potential in an
insulator, with or without a fixed background potential or a magnetic field, is
shown to be quantized. A similar phenomenon occurs in a quantum Hall system,
where the derivative of the electron density with respect to the magnetic field
is quantized in units of $e/h$, the inverse flux quantum. A number of other
interesting results may be unified under the notion of quantized density
response, including the quantization of adiabatic particle transport, the gap
labeling theorem and its generalization in the presence of a magnetic field.
