If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for $j_1,\ldots,j_d \in \mathbb{Z}$, then for each $q>1$ the generalised q-dimension, or generalised R\'enyi dimension of $\mu$ is defined to be the quantity $$d_q(\mu):=\lim_{r \to 0} \frac{\log \sum_{C \in \mathcal{M}_r} \mu(C)^q}{(q-1)\log r}$$ when this limit exists. I would like to know if $d_q(\mu)$ admits any simple upper bounds in terms of the dimension of the support of $\mu$. I have worked out some simple examples which suggest to me that the box dimension of the support might be such a bound, but it is not obvious to me how to prove this (perhaps because I do not have any intuition for the role of the factor $q-1$). Can anyone provide me with a reference or counterexample?
Thanks!