Large values of the additive energy in and

Combining Freiman's theorem with Balog–Szemerédi–Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of ${\mathbb{R}^d$ and ${\mathbb{Z}^d$.

Approximate groups and doubling metrics

We develop a version of Freĭman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates.Our work complements three other recent approaches to developing non-abelian versions of Freĭman's theorem by Breuillard and Green, Fisher, Katz and Peng, and Tao.

Freiman's Theorem in Finite Fields via Extremal Set Theory

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in $\F_2^n$: if $A \subseteq \F_2^n$ is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size $2^{2K + O(\sqrt{K}\log K)}|A|$; except for the $O(\sqrt{K} \log K)$ error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

A Note on Freĭman's Theorem in Vector Spaces

A famous result of Freĭman describes the sets A, of integers, for which |A+A| ≤ K|A|. In this short note we address the analogous question for subsets of vector spaces over $\mathbb{F}_2$. Specifically we show that if A is a subset of a vector space over $\mathbb{F}_2$ with |A+A| ≤ K|A| then A is contained in a coset of size at most 2O(K3/2 log K)|A|, which improves upon the previous best, due to Green and Ruzsa, of 2O(K2)|A|. A simple example shows that the size may need to be at least 2Ω(K)|A|.