The Polynomial Freiman-Ruzsa Conjecture

The original purpose of this repository is to hold a Lean4 formalization of the proof of the Polynomial Freiman-Ruzsa (PFR) conjecture of Katalin Marton (see also this blog post). The statement is as follows: if $A$ is a non-empty subset of ${\bf F}_2^n$ such that $|A+A| \leq K|A|$, then $A$ can be covered by at most $2K^{12}$ cosets of a subspace $H$ of ${\bf F}_2^n$ of cardinality at most $|A|$. The proof relies on the theory of Shannon entropy, so in particular development of the Shannon entropy inequalities was needed.

After the primary purpose of the project was completed, a second stage of the project developed several consequences of PFR, as well as an argument of Jyun-Jie Liao that reduced the exponent $12$ to $11$. This second stage has also been completed.

Currently, the project is obtaining an extension of PFR to other bounded torsion groups, as well as formalizing a further refinement of Jyun-Jie Liao that improves the exponent further to $9$.

• Discussion of the project on Zulip
• Blueprint of the proof
• Documentation of the methods
• A quick "tour" of the project
• Some example Lean code to illustrate the results in the project

Build the Lean files

To build the Lean files of this project, you need to have a working version of Lean. See the installation instructions (under Regular install).

To build the project, run lake exe cache get and then lake build.

Build the blueprint

To build the web version of the blueprint, you need a working LaTeX installation. Furthermore, you need some packages:

sudo apt install graphviz libgraphviz-dev
pip uninstall -y leanblueprint
pip install -r blueprint/requirements.txt

To actually build the blueprint, run

lake exe cache get
lake build
inv all

To view the web-version of the blueprint locally, run inv serve and navigate to http://localhost:8000/ in your favorite browser.

Or you can just run inv dev instead of inv all and inv serve, after each edit to the LaTeX, it will automatically rebuild the blueprint, you just need to refresh the web page to see the rendered result.

Note: If you have something wrong in your LaTeX file, and the LaTeX compilation fails, LaTeX will stuck and ask for commands, you'll need to type X then return to exit LaTeX, then fix the LaTeX error, and run inv dev again.

Moving material to mathlib

As the first two phases of the project are completed, we are currently working towards stabilising the new results and contributing them to mathlib.

Source reference

[GGMT]: https://arxiv.org/abs/2311.05762

[L] : https://arxiv.org/abs/2404.09639

[GGMT2]: https://arxiv.org/abs/2404.02244