Multifractal aspects of $\alpha$-expansions

In \cite{[NE]} we introduce $\alpha$-expansions a real numbers in $(0,1]$, given by $$\sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}$$ with $\alpha>1$ and $d_{i}\in\mathbb{N}$ and discuss ergodic theoretical and dimension theoretical aspects of this expansions. In this sequel we study mutifractal aspects of this expansions.