On distinct cross-ratios and related growth problems

Abstract

It is shown that for a finite set $A$ of four or more complex numbers, the cardinality of the set $C[A]$ of all cross-ratios generated by quadruples of pair-wise distinct elements of $A$ is $|C[A]|\gg |A|^{2+\frac{2}{11}}\log^{-\frac{6}{11}} |A|$ and without the logarithmic factor in the real case. The set $C=C[A]$ always grows under both addition and multiplication. The cross-ratio arises, in particular, in the study of the open question of the minimum number of triangle areas, with two vertices in a given non-collinear finite point set in the plane and the third one at the fixed origin. The above distinct cross-ratio bound implies a new lower bound for the latter question, and enables one to show growth of the set $\sin(A-A),\;A\subset \mathbb R/\pi\mathbb Z$ under multiplication. It seems reasonable to conjecture that more-fold product, as well as sum sets of this set or $C$ continue growing ad infinitum.