On the Structure of Hrushovski's Pseudoplanes Associated to Irrational Numbers (Model theoretic aspects of the notion of independence and dimension)

Abstract

Let α be an irrational number, and a/b a reduced fraction. Suppose 2/3 < α < a/b < 3/4 and b is sufficiently large. Let B be a canonical twig for a/ b and A the set of all leaves in B. Let p ∈ B be a good vertex of B over A. Let M be the generic structure for (K[f], <) where f is the Hrushovski's log-like function associated to a. Assume that B is a closed subset of M. Let D be the orbit of p over A in M. Then M = cl(D). Actually, we can prove this only assuming O < α < a/b < 1.