On point sets with many unit distances in few directions

We study the problem of the maximum number of unit distances among n points in the plane, under the additional restriction that we count only those unit distances that occur in a fixed set of k directions, taking the maximum over all sets of n points and all sets of k directions. We prove that, for fixed k and sufficiently large n > n(0)(k), the extremal sets are essentially sections of lattices, bounded by edges parallel to the k directions and of equal length.