https://github.com/YaelDillies/apap

Formalisation of the Kelley-Meka bound on Roth numbers
https://github.com/YaelDillies/apap

additive-combinatorics combinatorics lean4

Last synced: 3 months ago
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Formalisation of the Kelley-Meka bound on Roth numbers

• Host: GitHub
• URL: https://github.com/YaelDillies/apap
• Owner: YaelDillies
• License: apache-2.0
• Created: 2023-03-15T09:31:38.000Z (over 3 years ago)
• Default Branch: master
• Last Pushed: 2026-03-20T10:08:27.000Z (3 months ago)
• Last Synced: 2026-03-29T08:35:37.343Z (3 months ago)
• Topics: additive-combinatorics, combinatorics, lean4
• Language: Lean
• Homepage: https://yaeldillies.github.io/apap/
• Size: 1.59 MB
• Stars: 21
• Watchers: 3
• Forks: 8
• Open Issues: 2
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Arithmetic Progressions - Almost Periodicity

[![.github/workflows/push.yml](https://github.com/YaelDillies/apap/actions/workflows/push.yml/badge.svg)](https://github.com/YaelDillies/apap/actions/workflows/push.yml)

[![Gitpod Ready-to-Code](https://img.shields.io/badge/Gitpod-ready--to--code-blue?logo=gitpod)](https://gitpod.io/#https://github.com/YaelDillies/apap)

The purpose of this repository is to *digitise* some mathematical definitions, theorem statements

and theorem proofs. Digitisation, or formalisation, is a process where the source material,

typically a mathematical textbook or a pdf file or website or video, is transformed into definitions

in a target system consisting of a computer implementation of a logical theory (such as set theory

or type theory).

## The source

The definitions, theorems and proofs in this repository are taken from the exposition of Bloom and

Sisask on the Kelley-Meka bound on Roth numbers [2302.07211](https://arxiv.org/abs/2302.07211).

The main result is that there is some constant `c > 0` such that, if `A ⊆ {1, ..., N}` contains no

non-trivial arithmetic progression of length 3, then `|A| ≤ N/exp(c * (log n)^(1/12)))` for some

constant `c > 0`. This is an amazing improvement over previous bounds, which were all of the form

`N/(log n)^c` for some constant `c`.

## The target

The formal system which we are using as a target system is Lean's dependent type theory. Lean is a

project being developed at AWS and Microsoft Research by Leonardo de Moura and his team.

## Content of this project

This project currently contains about 3k lines of Lean code about the discrete (difference)

convolution, discrete Lp norms, discrete Fourier transform. It also contains proofs of a version of

almost periodicity and of a quantitative version of the Marcinkiewicz-Zygmund inequality.

Once finished, this project will contain two main results (here `R` is the Roth number, the maximum

size of a set without three term arithmetic progressions):

* The **finite field case**: A proof that `R(F_q^n) ≤ q ^ (n - c * n ^ (1/9))` for some constant

  `c`. This is worse than the Ellenberg-Gijswijt bound `R(F_q^n) ≤ q ^ (n - c * n)` which was

  formalised in [Dahmen, Hölzl, Lewis](https://drops.dagstuhl.de/opus/volltexte/2019/11070/). The

  goal here is therefore not to improve on the existing bound but instead demonstrate the

  probability and Fourier analysis techniques, whereas Ellenberg-Gijswijt used the polynomial

  method.

* The **integer case**: A proof that `R(n) ≤ N/exp(c * (log n)^(1/9)))`, using the same techniques

  as in the finite field case, except for the fact that we now use Bohr sets instead of subspaces.

  This bound is a slight improvement over the Kelley-Meka bound (with `1/12` as the exponent instead

  of `1/9`). It is due to Bloom and Sisask.

### Code organisation

The Lean code is contained in the directory `APAP`. The subdirectories are:

* `Mathlib`: Material missing from existing mathlib developments

* `Prereqs`: New developments to be integrated to mathlib

* `Physics`: The physical (as opposed to Fourier space) proof steps that are shared

  between the finite field cases and integer case

* `FiniteField`: The proof steps specific to the finite field case

* `Integer`: The proof steps specific to the integer case

## What next?

Almost periodicity is nowadays a standard tool in additive combinatorics. The version we formalised is sufficient for many applications. In particular, it gives one of the best known bounds on Freiman's theorem. As a side goal, we might tackle Freiman's theorem.

The discrete convolution/Lp norm/Fourier transform material belongs in mathlib and we hope to PR it there soon. Almost periodicity should similarly be upstreamed to mathlib given the numerous applications. The rest of the material might forever live in this repository.

On top of the new developments, there are many basic lemmas needed for this project that are currently missing from mathlib.

## Getting the project

To build the Lean files of this project, you need to have a working version of Lean.

See [the installation instructions](https://lean-lang.org/install/).

Alternatively, click on the button below to open an Ona workspace containing the project.

[![Open in Gitpod](https://gitpod.io/button/open-in-gitpod.svg)](https://gitpod.io/#https://github.com/YaelDillies/mean-fourier)

In either case, run `lake exe cache get` and then `lake build` to build the project.

## Build the blueprint

See instructions at https://github.com/PatrickMassot/leanblueprint/.

## Acknowledgements

Our project builds on mathlib. We must therefore thank its numerous contributors without whom this

project couldn't even have started.

Much of the project infrastructure has been adapted from

* [sphere eversion](https://leanprover-community.github.io/sphere-eversion/)

* [liquid tensor experiment](https://github.com/leanprover-community/liquid/)

* [unit fractions](https://github.com/b-mehta/unit-fractions/)

## Source reference

`[BS]` : https://arxiv.org/abs/2302.07211

[BS]: https://arxiv.org/abs/2302.07211