On a vector space analogue of Kneser’s theorem

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1-9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0EA<∞ and 0EB<∞. Then dim_EAB?dim_EA+dim_EB-dim_EH(AB), where AB is the E-space generated by {ab:a∈A,b∈B} and H(AB)={x∈K:xAB⊂AB}. The separability assumption was essential in the proof of this result. However, even without the separability assumption, no counterexample is known. The present paper shows that no counterexample can be found if dim_EA?5.