Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture

Joseph Harrison, Akshat Mudgal, Harry Schmidt

Abstract

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on arithmetic progressions in coordinates of elliptic curves, along with various other generalisations studied in the literature. We also prove a uniform Bourgain--Chang-type sum-product estimate for general $1$-dimensional algebraic groups $G$ over $\mathbb{C}$. Using these ideas, we provide an alternative solution to a problem of Bays--Breuillard. Furthermore, we show an Elekes--Szabó type result in the same setting for sets with small doubling, improving upon an earlier result of Bays--Breuillard when $G$ is not $\mathbb{G}_a$. Our power saving here can be shown to be quantitatively optimal. We use a combination of deep, classical results in Diophantine geometry due to David--Philippon, Laurent and Evertse--Schmidt--Schlickewei along with the recent breakthrough work on the weak Polynomial Freiman--Ruzsa conjecture over integers due to Gowers--Green--Manners--Tao.

Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture

Abstract

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on arithmetic progressions in coordinates of elliptic curves, along with various other generalisations studied in the literature. We also prove a uniform Bourgain--Chang-type sum-product estimate for general -dimensional algebraic groups over . Using these ideas, we provide an alternative solution to a problem of Bays--Breuillard. Furthermore, we show an Elekes--Szabó type result in the same setting for sets with small doubling, improving upon an earlier result of Bays--Breuillard when is not . Our power saving here can be shown to be quantitatively optimal. We use a combination of deep, classical results in Diophantine geometry due to David--Philippon, Laurent and Evertse--Schmidt--Schlickewei along with the recent breakthrough work on the weak Polynomial Freiman--Ruzsa conjecture over integers due to Gowers--Green--Manners--Tao.
Paper Structure (16 sections, 28 theorems, 127 equations)

This paper contains 16 sections, 28 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Further Applications
  3. Generalised Bremner
  4. Sum-product phenomenon
  5. Elekes--Szabó
  6. Setup
  7. Algebraic groups
  8. Degrees
  9. Correspondences
  10. Mordell--Lang and $S$-unit equations.
  11. Projecting cartesian products
  12. Correspondences and cosets
  13. Freiman--type structural theorems
  14. Proofs of main results
  15. Diophantine equations
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is an effectively computable constant $C \geq 1$ with the following property. Let $E$ be an elliptic curve in Weierstrass form and let $r$ be the rank of $E(\mathbb{Q})$. Let $X = \{x(P): P \in E(\mathbb{Q})\}$ and $Y = \{y(P): P \in E(\mathbb{Q})\}$. Let $A$ be either an arithmetic progression, a geometric progression or a set of the form with $u,d\in \mathbb{Q}$, and $l \in \mathbb{N}$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Example 1.6
  • Definition 1.7
  • Theorem 1.8
  • Example 1.9
  • Theorem 2.1
  • ...and 48 more