We settle the Polynomial Freiman--Ruzsa (PFR/Marton) conjecture for the integers and for cyclic groups. More precisely, we show that if $A$ is a finite subset of $\mathbb{Z}$ or $\mathbb{Z}/N\mathbb{Z}$ with $|A+A| \le K|A|$, then there is a subgroup $H$ of index at most $K^{O(1)}$ such that $A$ is contained in at most $K^{O(1)}$ cosets of $H$. The proof is based on a new spectral stability dichotomy for the $L^4$ Fourier mass of $\mathbf{1}_A$: either this mass is concentrated on a span of size $K^{O(1)}$, or, after passing to a quotient of codimension $K^{O(1)}$, the doubling constant of the image of $A$ decreases by a definite power of $K$. Using Freiman modeling we transfer this dichotomy to cyclic groups, obtain polynomial Bogolyubov-type bounds, and deduce Marton's conjecture in $\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$. As a corollary, we also recover and extend the finite-field formulation of Marton's conjecture: in odd characteristic we obtain a direct spectral proof, and together with the characteristic-2 result of Green, Gowers, Manners, and Tao this yields a complete resolution of the conjecture for all finite fields. For context beyond finite fields, we recall their theorem for abelian groups of bounded exponent.
The Polynomial Freiman-Ruzsa conjecture is one of the central open problems in additive combinatorics. If true, it would give tight quantitative bounds relating combinatorial and algebraic notions of approximate subgroups. In this note, we restrict our attention to subsets of Euclidean space. In this regime, the original conjecture considers approximate algebraic subgroups as the set of lattice points in a convex body. Green asked in 2007 whether this can be simplified to a generalized arithmetic progression, while not losing more than a polynomial factor in the underlying parameters. We give …
Recently Breuillard and Tointon showed that one reasonable formulation of the polynomial Freiman-Ruzsa conjecture fails for nonabelian groups. We improve and simplify their construction.
A non-quantitative version of the Freiman-Ruzsa theorem is obtained for finite stable sets with small tripling in arbitrary groups, as well as for (finite) weakly normal subsets in abelian groups.
Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman--Ruzsa theorem asserts that if |A+A| < K|A| then A is contained in a coset of a subgroup of G of size at most r^{K^4}K^2|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to r^{CK}|A| for some absolute constant C >= 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion.
In this paper, we study the linear structure of sets $A \subset \mathbb{F}_2^n$ with doubling constant $σ(A)<2$, where $σ(A):=\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in a small affine subspace. We also show that $A$ can be covered by at most four shifts of some subspace $V$ with $|V|\leq |A|$. Finally, we classify all binary sets with small doubling constant.
Let $p$ be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p$ can be stated as follows. If $ϕ: \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ is a function such that $ϕ(x+y) - ϕ(x) - ϕ(y)$ takes values in some set $S$, then there is a linear map $\tildeϕ : \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ with the property that $ϕ- \tildeϕ$ takes at most $|S|^{O(1)}$ values. A strong variant of this conjecture states that, in fact, there is a linear map $\tildeϕ$ such that $ϕ- \tildeϕ$ takes values in $tS$ for some constant $t$. In this note, we discuss a counterexamp…
We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Freĭman dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of as a major quantitative improvement on Freĭman's dimension lemma, or as a "weak" Freĭman--Ruzsa theorem with almost polynomial bounds. The methods used, centered around an "additive energy increment strategy", differ from the usual tools in this area and may have further potential. Most of our argument takes place over $\mathbb{F}_2^n$, which is itself curious. T…
University of Cambridge（剑桥大学）；Centrum Wiskunde & Informatica (CWI)（数学与计算机科学中心）；Leiden University（莱顿大学）；IBM Research, Silicon Valley Lab（IBM硅谷实验室）；IBM Research, Cambridge（IBM剑桥研究中心）
Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in 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(K^{46}) 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 i…
The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$. The estimation of the constants $d$ and $c'$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.
We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem.
Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. We extend these results further to solvable groups of bounded derived length, in which the coset progressions are replaced by the more complicated notion of a "coset nilprogression". As one consequence of this result, any subset of such a solvable group of sm…
We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their centre.
Let $G = (G,+)$ be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others characterises those finite sets $A$ for which $A+A$ is small. In this paper we establish analogous results in which the finite set $A \subset G$ is replaced by a discrete random variable $X$ taking values in $G$, and the cardinality $|A|$ is replaced by the Shannon entropy $\mathrm{Ent}(X)$. In parti…
Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (l…
