The addition of binary lists has been a well-studied problem in mathematics. If the sets are random, the resulting sums are typically all different. However, sets that are not random and not subgroups have been difficult to study. The Freiman-Ruzsa conjecture proposed a polynomial version of this problem, but a proof has been elusive. Recently, mathematicians Ben Green, Ryan Manners, Terence Tao, and Tim Gowers discovered that by measuring the size of a set using entropy rather than the number of elements, they could make progress on the conjecture. After months of collaboration, they posted their paper, which provides a proof for the polynomial Freiman-Ruzsa conjecture. The proof was then formalized with the help of 25 volunteers using GitHub and Blueprint. The result is a breakthrough in the field of additive combinatorics and provides new insights into the structure of sets with small sumsets.
https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/