  A multi-resolution quantizer is a sequence of quantizers where the output of
a coarser quantizer can be deduced from the output of a finer quantizer. In
this paper, we propose an asymptotically scale-invariant multi-resolution
quantizer, which performs uniformly across any choice of average quantization
step, when the length of the range of input numbers is large. Scale invariance
is especially useful in worst case or adversarial settings, ensuring that the
performance of the quantizer would not be affected greatly by small changes of
storage or error requirements. We also show that the proposed quantizer
achieves a tradeoff between rate and error that is arbitrarily close to the
optimum.
