Quantization dimensions of negative order

TL;DR

This paper extends the concept of quantization dimensions to include negative orders for probability measures, providing bounds, conditions, and asymptotics, and addresses an open question about geometric mean error.

Contribution

It introduces a framework for defining and analyzing quantization dimensions of negative order measures, including bounds and asymptotics, extending prior work.

Findings

Established bounds for negative order quantization dimensions

Provided conditions for existence of these dimensions

Solved an open problem on geometric mean error asymptotics

Abstract

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order $r$, including negative values of $r$. To this end, we use the concept of partition functions, which generalizes the idea of the $L^{q}$-spectrum and in this way naturally extends the work in [M. Kesseb\"ohmer, A. Niemann, and S. Zhu. Quantization dimensions of probability measures via R\'enyi dimensions. Trans. Amer. Math. Soc. 376.7 (2023)]. In particular, we provide natural fractal geometric bounds as well as easily verifiable necessary conditions for the existence of the quantization dimensions. The exact asymptotics of the quantization error of negative order for absolutely continuous measures are stated, whereby an open question from [S. Graf, H. Luschgy. Math. Proc. Cambridge Philos. Soc. 136, 3 (2004)] regarding the geometric mean error is also answered in the affirmative.