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38 pages, Latex revised version: draft formulae indicator deleted, one reference added, typos corrected, few minor text modificationsNon-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formulæ and compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations

On the finite size corrections of anti-ferromagnetic anomalous dimensions in Script N = 4 SYM