How projections affect the dimension spectrum of fractal measures

We introduce a new potential-theoretic definition of the dimension spectrum D-q of a probability measure for q > I and explain its relation to prior definitions. We apply this definition to prove that if 1 < q less than or equal to 2 and mu is a Borel probability measure with compact support in R-n, then under almost every linear transformation from R-n to R-m, the q-dimension of the image of mu is min(m, D-q(mu)); in particular, the q-dimension of mu is preserved provided m greater than or equal to D-q(mu). We also present results on the preservation of information dimension D-I and pointwise dimension. Finally, for 0 less than or equal to q < 1 and q > 2 we give examples for which D-q is not preserved by any linear transformation into R-m. All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.