Counting points in thin sets: A survey

TL;DR

This survey reviews recent advances in counting rational points of bounded height in thin sets, highlighting resolved questions, open problems, and the complexity of uniform bounds especially for type II thin sets.

Contribution

It provides a comprehensive overview of progress on counting points in thin sets, clarifies the state of uniform bounds, and discusses open questions and counterexamples.

Findings

Progress on counting points in four types of thin sets.

Existence of uniform bounds in type I thin sets.

Counterexamples indicating complexity in uniformity for type II thin sets.

Abstract

In the 1980's Serre asked how many points of bounded height can lie in a thin set. This has motivated significant research ever since, culminating in a series of recent breakthroughs. It is a good time to take stock of the central questions that have been resolved, and also to highlight remaining open questions. First, we survey recent progress on counting points of bounded height in the four types of thin sets, according to the projective/affine and type I/type II designations. Second, we turn to questions of uniformity. Famously, in the setting of type I thin sets, the best-known upper bound for the number of points of bounded height is independent of the maximum size, say $\|F\|$, of the coefficients of the polynomials that define the thin set; such an upper bound is called uniform. A uniform upper bound in the setting of type II thin sets is not known. For type II thin sets, we explore the dependence on $\|F\|$ via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.