Counting Rational Points on Foliations via Pila-Wilkie Extensions — E8 Intelligence Research

FINDING: Pila-Wilkie counting is being extended from algebraic/transcendental sets to foliations and rigid-analytic sets, yielding sub-polynomial bounds on rational points. | MATH: The core is the Pila-Wilkie theorem: for a set definable in an o-minimal structure, the number of rational points of height ≤ H in the transcendental part is \\(O(H^\\epsilon)\\) for all ε>0. Binyamini's extension: for a leaf \\(L\\) of a foliation on an algebraic variety \\(V\\) over a number field, the counting function \\(N(L, H)\\) (rational points of height ≤ H on a compact piece of L) is also \\(O(H^\\epsilon)\\). The rigid-analytic analog (arXiv:2203.10530) gives the same sub-polynomial bound for \\(\\mathbb{Q}_p\\)-analytic sets and for \\(\\mathbb{F}_q((t))\\)-analytic sets, counting rational functions of degree ≤ d as \\(O(d^\\epsilon)\\). | CONNECTION: The sub-polynomial growth \\(H^\\epsilon\\) is the analytic signature of *transcendence* — it contrasts with polynomial growth \\(H^d\\) for algebraic sets. The threshold be Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com