On the resolution of Marton's conjecture from additive combinatorics

In this article, Briët delves into the history and scope of Marton’s conjecture, which is also known as the polynomial Freĭman-Ruzsa conjecture. This conjecture concerns set coverings by cosets of subgroups in case of small doubling in finite-field setting. Already in 1927, Van der Waerden published an article about arithmetic progressions (APs) in this magazine. Via the conjecture by Erdős en Turán, an ergodic version by Furstenberg and proofs constructed via Fourier-analytic techniques and ergodic theory, Briët arrives at the proof of Marton’s conjecture by Gowers, Green, Manners and Tao in 2023. This proof was achieved using information-theoretic tools, in particular, the entropic Ruzsa distance. Briët concludes with the importance of equivalent formulations for various fields and a brief outline.