2017/05/04 by Misha Rudnev, Rudnev, Misha
Computer Science · Mathematics · #11B75 #68R05 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1705.01830
openalex publication_date 2017/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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]|≫ |A|2+(2)/(11)log-(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⊂ \mathbb R/π\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.