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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!

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After about 24 hours I understood how this estimate works, so I'm posting it in case anyone else has the same problem. Let $N(r):=\#\{C \in \mathcal{M}_r \colon \mu(C)>0\}$. We note that if $\sum_{i=1}^np_i=1$, $0 \leq p_i \leq 1$, $q>1$ then $$\sum_{i=1}^n p_i^q \geq n^{1-q},$$ so for $0<r<1$ $$\frac{\log \sum_{C \in \mathcal{M}_r}\mu(C)^q}{(q-1)\log r} \leq \frac{\log N(r)}{\log r}$$ and the limit (superior) of the latter as $r \to 0$ is simply the (upper) box dimension of the support of $\mu$.

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