Quantization dimensions of negative order

2024/05/22 by Marc Keßeböhmer, Kesseböhmer, Marc, Aljoscha Niemann +1
Physics and Astronomy · #28A80 #60E05 #62E20 #94A12 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Optical and Acousto-Optic Technologies #Optimization and Control (math.OC) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2405.13387

openalex publication_date 2024/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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. 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.

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