Bases in some additive groups and the Erdős–Turán conjecture

We show that the analogue of the Erdős–Turán conjecture, for the number of representations by a basis of order two of an additive semi-group, does not hold in a variety of additive groups derived from those of certain fields. This is done by explicitly constructing some bases for which we estimate the maximal number of representations of the elements of the group as a sum of two elements from the given basis.