Exact Redundancy for Symmetric Rate-Distortion

For variable-length coding with an almost-sure distortion constraint, Zhang et al. show that for discrete sources the redundancy is upper bounded by $\log n/n$ and lower bounded (in most cases) by $\log n/(2n)$, ignoring lower order terms. For a uniform source with a distortion measure satisfying certain symmetry conditions, we show that $\log n/(2n)$ is achievable and that this cannot be improved even if one relaxes the distortion constraint to be in expectation rather than with probability one.