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Quantization dimensions of negative order

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. Kesseböhmer, A. Niemann, and S. Zhu. Quantization dimensions of probability measures via Rényi 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.