Quantization dimensions of negative order

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 employ the concept of partition functions, which generalises the notion of the $L^q$ -spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.