Comparison Based Sorting Run-time with respect to Total Number of Bits of Input
Assume, as is often the case, that all elements in the list have width $\Theta(\log n)$. This means that $w = \Theta(\log n)$ and so $x = nw = \Theta(n\log n)$. You can check that $n = \Theta(x/\log x)$, and this allows you to convert running times stated as a function of $n$ to running times stated as a function of $x$. For example, if the running time of some algorithm is $\Theta(n^2)$ as a function of $n$, then its running time as a function of $x$ is $\Theta(x^2/\log^2 x)$.