Points of bounded height on oscillatory sets

TL;DR

This paper demonstrates that certain transcendental curves, including logarithmic spirals and solutions to specific Euler equations, contain very few rational points of bounded height under certain algebraic and parametrization conditions.

Contribution

It establishes a new framework for bounding rational points on transcendental curves using algebraic and differentiable function parametrizations.

Findings

Transcendental curves with specific parametrizations have finitely many rational points of bounded height.

Examples include logarithmic spirals and solutions to Euler differential equations.

The results extend previous work on rational points to a broader class of transcendental curves.

Abstract

We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by functions belonging to a specified algebra of infinitely differentiable functions. Examples of such curves include logarithmic spirals and solutions to Euler equations $x^2y''+xy'+cy=0$ with $c>0$.