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We study the leading quantum string correction to the dressing phase in theasymptotic Bethe Ansatz system for superstring in AdS_3 x S^3 x T^4 supportedby RR flux. We find that the phase should be different from the BES phaseappearing in the AdS_5 x S^5 case. We use the simplest example of a rigidcircular string with two equal spins in S^3 and also consider the generalapproach based on the algebraic curve description. We also discuss the case ofthe AdS_3 x S^3 x S^3 x S^1 theory and find the dependence of the 1-loopcorrection to the effective string tension function h(\lambda) (expected toenter the magnon dispersion relation) on the parameters alpha related to theratio of the two 3-sphere radii. This correction vanishes in the AdS_3 x S^3 xT^4 case.

Quantum corrections to spinning superstrings in AdS_3 x S^3 x M^4: determining the dressing phase