Abstract
New lower bounds involving sum, difference, product, and ratio sets of a set $A\subset {\mathbb C}$ are given. The estimates involving the sum set match, up to constants, the state-of-the-art estimates, proven by Solymosi for the reals and are obtained by generalizing his approach to the complex plane. The bounds involving the difference set improve the currently best known ones, also due to Solymosi, in both the real and complex cases by means of combining the Szemeredi--Trotter theorem with an arithmetic combinatorics technique.
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