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Counting points in thin sets: A survey

Dante Bonolis, Lillian B. Pierce, Katharine Woo

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.

Counting points in thin sets: A survey

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 , 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 via several strategies, and construct counterexamples that suggest the question of uniformity is quite subtle in the setting of type II thin sets.
Paper Structure (36 sections, 31 theorems, 169 equations)

This paper contains 36 sections, 31 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Thin sets of type I and type II: definitions
  4. The counting problems
  5. Questions of dependence and uniformity
  6. Notation
  7. Preliminaries: motivation for thin sets of type I and type II
  8. Terminology
  9. The two types
  10. Proof of Lemma \ref{['lemma_thin_set_two_types']}
  11. Proof of Lemma \ref{['lemma_typeII_poly_interp']}
  12. Proof of Lemma \ref{['lemma_affine_version_is_thin']}: simple relation between projective and affine thin sets
  13. Type I
  14. Type II
  15. Survey of type I thin sets and Dimension Growth
  16. ...and 21 more sections

Key Result

Lemma 1.1

Every thin set in $\mathbb{P}^{n-1}(\mathbb{Q})$ (respectively in $\mathbb{A}^{n}(\mathbb{Q})$) is a finite union of thin sets of type I and type II.

Theorems & Definitions (63)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.3: Baseline upper bound
  • Lemma 1.4
  • Conjecture 1.5: Thin set, projective
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2: EGAIV_part3
  • Lemma 2.3
  • ...and 53 more