Geometric mean quantization via adaptive approximation

Abstract

Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quantization dimensions of $\nu$ are the lower and upper limits of $\log(1/t)/\mathcal{L}_{\nu}(t)$. The dimension exists precisely when $(q-1)\sum_{Q}\nu(Q)^{q}$, summed over all dyadic cubes, converges as $q\downarrow1$, and it is then determined by this limit. The mass-threshold formula yields harmonic integral bounds in terms of the local dimensions and encloses entropy and quantization dimensions in a common spectral interval. Convergence in law of the local information rates is equivalent to convergence of the rescaled spectra in a window of width $1/k$ around $q=1$; the two dimensions are then the arithmetic and the harmonic mean of the limit law, and we quantify their difference by sharp bounds and variance identities. Without any convergence assumption, vanishing threshold variance still forces equality of the corresponding lower and upper dimensions. Bernoulli mixtures realise every local-dimension law with compact support in $(0,1]$, and a regime-switching example separates convergence in law from almost-everywhere convergence.