I am having difficulty computing the following formula for the Hausdorff Dimension for a type of Moran set called a partial homogenous Cantor set.
$\displaystyle\liminf_{k\rightarrow\infty}\displaystyle\frac{\log n_1...n_k}{-\log c_1...c_kc_{k+1}n_{k+1}}$
Where $n_1=2, n_{k+1}=[((n_1...n_k)^{\frac{1}{r}-\frac{1}{s}})^\frac{s}{1-s})]; c_k=n_k^{-\frac{1}{s}}$. Note that $n_k\in\mathbb{N}$ and $c_k\in\mathbb{R}_+$.
My purpose of performing this calculation is to check that the above formula is $<s$ for $r,s\in(0,1)$ as is claimed in this paper.
Write $n_1\cdots n_k=m_k$. Then your $\liminf$ is $$ \liminf_{k\to\infty}\frac{\log m_k}{-\log(m_k^{-1/s}n_{k+1}^{1-1/s})}= \liminf_{k\to\infty}\frac{\log m_k}{-\log(m_k^{-1/s}[m_k^{(1/r-1/s)s/(1-s)}]^{(s-1)/s})}. $$ The integer part is at least $m_k^{1/r-1/s}$ and less than $(2m_k^{(1/r-1/s)s/(1-s)})^{(s-1)/s})=2^{(s-1)/s}m_k^{1/r-1/s}$. Therefore, your $\liminf$ is $$ \frac{\log m_k}{-\log(m_k^{-1/s}m_k^{1/r-1/s})}=\frac{1}{-1/r+2/s} $$ (since $m_k$ is at least $2$, which ensures that the extra $2^{(s-1)/s}$ disappears).
I must say that I find curious that you mention nothing about the choice of $r$ and $s$ (which is not discussed in the paper that you mention). Of course, it would be nice that $s<2r$ but this probably depends on your specific construction.