On distinct cross-ratios and related growth problems

TL;DR

This paper establishes new lower bounds on the size of cross-ratio sets generated by finite complex sets, demonstrating their growth under addition and multiplication, with implications for geometric problems like triangle area counts.

Contribution

It provides the first nontrivial lower bounds on the size of cross-ratio sets for finite sets, showing their growth and applications to geometric and trigonometric problems.

Findings

Lower bounds on |C[A]| for complex sets, with explicit growth rates.

Cross-ratio sets grow under addition and multiplication.

Implications for minimum triangle areas and sine set growth.

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.