  The spectrum of anomalous dimensions of twist sl(2) operators in N=4 SYM has
an intriguing feature in low twist 2 or 3. The anomalous dimension of the
lowest state, dual a folded string on AdS_5 X S^5, can be computed by Bethe
Ansatz at 3, 4 loops respectively as a simple closed function of the Lorentz
spin. This feature is apparently lost at higher twist. We propose sum rules for
the excited anomalous dimensions where closed expressions can still be
provided, even at higher twist. We present several explicit three loop
examples. Many structural regularities can be observed leading to closed
expressions which depend parametrically both on the spin and the twist. They
allow to compute the subleading term in the logarithmic large spin expansion of
the sum rules as a compact simple function of the twist, in analogy with the
recent results by Freyhult, Rej and Staudacher in arXiv:0712.2743 [hep-th].
