Points of bounded height on oscillatory sets

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 $x^2{y}^{\prime \prime }+x{y}^{\prime }+cy=0$ with $c>0$.