On the sumset of binary recurrence sequences

Answering a question of I. Z. Ruzsa, A. Bérczes [1] gave a complete description of the restricted sumset of geometric progressions having positive real quotient. In this connection, it is natural to ask whether it is possible to give a similar description of the restricted sumset of binary recurrence sequences? The present paper answers the question for Lucas sequences. Several results on sumsets of various kind of sets are available in the literature. For such results we refer to [6], [4] and the references given there. However, since the results of the present paper are not much connected to those results, and the techniques of the proofs are also quite different, we omit to mention them explicitly. Recall that a Lucas sequence is a binary recurrence sequence given by