New lower bounds involving sum, difference, product, and ratio sets for a set A ⊂ C are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising his approach to the complex plane. The bounds involving the difference set are slightly weaker. They improve on the best known onesalso due to Solymosi, by means of combining the use of the Szemerédi-Trotter theorem with an arithmetic combinatorics technique.
Bibliographical note19pp. This is a new extended version, accepted for publication to SIAM J. Disc. Math. Note: the earlier homonymous preprint arXiv_math: 1111.4977 of the Second Author contained weaker estimates involving the sum-set. The present estimate for the sum-set was erroneously claimed in arXiv:0812.1454