  In the high spin limit the minimal anomalous dimension of (fixed) twist
operators in the $sl(2)$ sector of planar ${\cal N}=4$ Super Yang-Mills theory
expands as $\gamma(g,s,L)=f(g) \ln s + f_{sl}(g,L) + \sum \limits_{n=1}^\infty
\gamma^{(n)}(g,L) (\ln s)^{-n} + ... $. We find that the sub-logarithmic
contribution $\gamma^{(n)}(g,L) $ is governed by a linear integral equation,
depending on the solution of the linear integral equations appearing at the
steps $n'\leq n-3$. We work out this recursive procedure and determine
explicitly $\gamma^{(n)}(g,L) $ (in particular $\gamma^{(1)}(g,L)=0$ and
$\gamma^{(n)}(g,2)=\gamma^{(n)}(g,3)=0$). Furthermore, we connect the
$\gamma^{(n)}(g,L) $ (for finite $L$) to the generalised scaling functions,
$f^{(r)}_n(g)$, appearing in the limit of large twist $L\sim\ln s$. Finally, we
provide the first orders of weak and strong coupling for the first
$\gamma^{(n)}(g,L)$ (and hence $f^{(r)}_n(g)$).
