An intermediate-value theorem for the upper quantization dimension

Let μ be a Borel probability measure on Rd with compact support and the upper quantization dimension of μ of order r. We prove, that for every , there exists a Borel probability measure ν with ν≪μ such that . In addition, we give an example to show that the above intermediate-value property may fail in the open interval . Thus we get a complete description of the dimension set .