I'm trying to prove exercise 8.5 from 'The geometry of fractal sets' by Falconer:
Suppose that the open set condition holds for the contractions $\{\psi_j\}^m_1$ on $\mathbb{R}^n$. Suppose further that for each j and all $x, y\,$: $$q_j|x-y|\leq |\psi_j(x)-\psi_j(y)|\leq r_j|x-y|$$ If E is the invariant set associated with these contractions adapt the proof of Theorem 8.6 to show that $s'\leq dim \, E\leq t\;\,$ where $\;s'=s-\log(\max(r_j/q_j))/\log(\min(1/q_j))\;$ and where $\;\sum_1^mq_j^s=1=\sum_1^mr_j^s\,$.
If the open set $V$ for which the open set condition holds is contained in a ball of radius $c_2$ I think I've proved that the set $V_{j_1 \dots j_k}=\psi_{j_1}\circ\dots\psi_{j_k}(V)$ is contained in a ball of radius $$c_2\rho^{1-\frac{\log(\,\max(r_j/q_j)\,)}{\log(\,\min(1/q_j)\,)}}$$ where $\rho$ is the one in Theorem 8.5, (which means $\min(q_j)\rho\leq q_{j_1}\dots q_{j_k}<\rho$ holds for the sequences we consider).
The problem is that when I compare the volume of the balls, the exponent gets multiplied by $n$, so in the end I get the following lower bound for the Hausdorff dimension: $$s-n\,\frac{\log(\max(r_j/q_j))}{\log(\min(1/q_j))}\leq dim \,E$$ which is worse than the one in the exercise. Am I doing something wrong or is it a misprint in the book?