This post continues our series of online seminars devoted to accessible presentations of some of the most significant mathematical theorems of the 21st century. I am delighted to report that, on April 15, Prof. Harald Helfgott delivered an accessible lecture on growth, expansion, and escape in groups. If you missed the talk, the recording is now available on YouTube.
The next lecture in the series will take place on Prof. Ben Green (University of Oxford, UK) will speak on the resolution of the polynomial Freiman–Ruzsa conjecture.
To join the talk, simply click here at the scheduled time; no registration is required. For a full list of upcoming seminars, please visit https://th21.le.ac.uk/next-talks/.
For announcements of other talks, as well as descriptions of recent major mathematical breakthroughs, see the full list of posts on this blog.
Prof. Ben Green will speak about product theorems in groups.
For sets of integers and , write If is an arithmetic progression, then . A deep theorem of Freiman from 1973 (Freiman 1973) describes all possible examples of sets such that for fixed . An important line of research is to obtain analogues of Freiman’s theorem for groups other than .
Let be a subset of a group with operation . In which cases can one have for fixed ? An obvious example is when is a finite subgroup of , in which case . The same equality holds if is a coset of a finite subgroup , that is, for some . More generally, if is a subset of a subgroup such that , then implies . Similarly, if can be covered by cosets of , then can be covered by at most cosets, so Thus this quantity is bounded above by whenever . A natural question is whether one can reverse this reasoning: starting from the assumption , can one conclude that is contained in a subgroup of size at most , or at least that can be covered by cosets of such a subgroup?
In 1999, Ruzsa (Ruzsa 1999) proved that this is indeed the case for certain groups. More precisely, let be an integer, let be an abelian group in which every element has order at most , and let be a finite set such that . Then is contained in a subgroup of satisfying where . Ruzsa also cited a conjecture, which he attributed to Marton, asserting that under the same assumptions there should exist a subgroup of such that and is contained in the union of at most cosets of , where the constant depends only on . This statement became widely known as the polynomial Freiman–Ruzsa conjecture, and it has been described as the “holy grail of additive combinatorics.”
An important family of groups to which the conjecture applies is , the group of binary strings of length with componentwise addition modulo . This is an abelian group in which every element has order . In 2025, Gowers, Green, Manners, and Tao (Gowers et al. 2025) proved the polynomial Freiman–Ruzsa conjecture for this family of groups.
Liao (Liao 2024) later observed that Theorem remains true with . In the opposite direction, it is known (Green and Tao 2009) that one must have .
Theorem has many corollaries and equivalent formulations. In particular, it was observed in (Green et al. 2025) that it implies the so-called “weak polynomial Freiman–Ruzsa conjecture over ”. The latter predicts that if is an integer and is a finite subset of satisfying , then there exists a subset with whose dimension is at most . Theorem implies that this is true, in fact with and .
In later work (Gowers et al. 2024), the same authors generalized Theorem to arbitrary abelian groups of bounded torsion, thereby establishing the polynomial Freiman–Ruzsa conjecture in full generality in that setting. Recall that an abelian group has torsion if for all . The main result of (Gowers et al. 2024) states that if is an abelian group of torsion and is a finite non-empty set satisfying , then can be covered by at most cosets of some subgroup of of size at most . A canonical family of examples is given by for a fixed prime , where every such group has torsion .
For further progress toward a version of the polynomial Freiman–Ruzsa conjecture for arbitrary abelian groups, with no restriction on torsion, see (Raghavan 2025).