An intermediate-value theorem for the upper quantization dimension

AbstractLet μ be a Borel probability measure on Rd with compact support and D¯r(μ) the upper quantization dimension of μ of order r. We prove, that for every t∈(dimp∗μ,dim¯B∗μ], there exists a Borel probability measure ν with ν≪μ such that D¯r(ν)=dim¯B∗ν=t. In addition, we give an example to show that the above intermediate-value property may fail in the open interval (dimpμ,dimp∗μ). Thus we get a complete description of the dimension set {D¯r(ν):ν(Rd)=1,ν≪μ}