Marton-Type Freiman–Ruzsa Theorems

• Marton-type Freiman–Ruzsa theorems characterize small-doubling sets as those that can be covered by a bounded number of cosets with polynomial dependency on the doubling constant.
• They utilize a combination of entropic iteration and Fourier-analytic stability techniques to derive explicit structural bounds in groups like F₂ⁿ and Fₚⁿ.
• Recent advances improve exponent bounds (e.g., reducing the constant in F₂ⁿ from 12 to 9) and extend the framework to bounded torsion and distributional settings.

A Marton-type Freiman–Ruzsa theorem (often abbreviated as a Marton-type FR theorem) refers to structural results in additive combinatorics quantifying the containment or covering of small-doubling sets within large algebraic structures, most notably cosets of subgroups or progressions, with bounds that are polynomial (rather than exponential) in the doubling constant. These theorems originate from Katalin Marton's formulation of the so-called "polynomial Freiman–Ruzsa conjecture," which posits such strong polynomial bounds. In recent years, significant advances have resolved Marton's conjecture in broad settings, with characteristic-vector-space and bounded-torsion Abelian groups being central cases.

1. Definition and Scope

Given an abelian group $G$ and $A \subseteq G$, $A$ has doubling at most $K$ if $|A+A| \leq K |A|$. The classical Freiman–Ruzsa theorems provide that such $A$ must be "structured": in $\mathbb{F}_2^n$ or more generally in bounded-torsion groups, $A$ can be contained within or covered by boundedly many cosets of a subgroup $H \le G$ with $|H| \lesssim |A|$. A Marton-type Freiman–Ruzsa theorem is one in which the covering number and $|H|/|A|$ are bounded by polynomials (in $K$ and in the group’s torsion parameter), as opposed to the exponential dependencies in classical results.

For instance, in $G = \mathbb{F}_2^n$, Marton's conjecture (proved in (Gowers et al., 2023), with exponent improved in (Liao, 15 Apr 2024)) posits that any $A$ with $|A+A| \le K|A|$ can be covered by at most $2K^{C}$ cosets of a subgroup $H$ with $|H| \le |A|$, where $C$ is an absolute constant.

2. Main Results and Exponents

$\mathbb{F}_2^n$ and Characteristic 2

The polynomial Freiman–Ruzsa theorem in $G = \mathbb{F}_2^n$ states: If $A \subseteq \mathbb{F}_2^n$ satisfies $|A+A| \le K|A|$, then there exists a subgroup $H \le \mathbb{F}_2^n$ with $|H|\le|A|$ such that $A$ can be covered by at most $2K^C$ cosets of $H$, for $C$ absolute. The latest improvement establishes $C = 9$ (Liao, 15 Apr 2024), down from $C=12$ in the original proof of Gowers–Green–Manners–Tao (Gowers et al., 2023).

Odd Characteristic and General Torsion

For vector spaces over odd prime fields $\mathbb{F}_p^n$ ($p$ odd), recent results (Moghadam, 4 Dec 2025, Even-Zohar et al., 2012) establish analogues: $A\subset\mathbb{F}_p^n$ with $|A+A|\le K|A|$ is covered by $K^{O(1)}$ cosets of a subgroup $H$, $|H| \le K^{O(1)}|A|$. For arbitrary bounded torsion Abelian groups $G$ of exponent $m$, the covering number is at most $(2K)^{O(m^3)}$, with $|H|\le|A|$ (Gowers et al., 2 Apr 2024).

Entropic Structural Variant

An important Marton-type FR theorem refers to an entropic (distributional) setting: If $A$ is such that the Shannon entropy of $U_A + U_A$ (where $U_A$ is the uniform measure on $A$) is only slightly larger than $H(U_A)$, i.e., $H(U_A + U_A) - H(U_A) \leq \eta$, then $A$ differs from a coset $H + x$ by only $O(\eta \log(1/\eta)|H|)$ elements (Karam, 16 Jun 2024).

3. Proof Strategies: Entropic and Fourier-Analytic Methods

Marton-type FR theorems in recent literature exploit a deep connection between combinatorial structure and entropy, with variants also relying on Fourier-analytic stability and covering arguments.

Entropic Iteration and Covering

The core innovation involves the entropic Ruzsa distance, defined for $\mathbb{F}_2^n$-valued random variables $X, Y$ as

$d(X;Y) = H(X'-Y') - \frac{1}{2}(H(X) + H(Y)),$

where $X', Y'$ are independent copies. The main inequality asserts that for such $X,Y$ with sufficiently small $d(X;Y)$, there exists a subgroup $V$ such that $d(X;U_V) + d(Y;U_V) \leq C' d(X;Y)$ for absolute $C'$.

The entropic Marton PFR is then derived by an iteration involving “sums” and “fibres” (conditional random variables), tracking potential functions $\tau_A, \tau_B$ and employing entropic analogues of the Balog–Szemerédi–Gowers lemma. Improvements in the endgame accounting yield tighter exponents: the latest advancement reduces the exponent for $C'$ from 11 to 10, and hence for the covering number from $K^{12}$ to $K^{9}$ in $\mathbb{F}_2^n$ (Liao, 15 Apr 2024).

Fourier-Analytic Stability

In $\mathbb{F}_p^n$ for odd primes, a Polynomial Stability Lemma (PSL) provides a dichotomy: either a large proportion of the $L^4$ Fourier mass of $1_A$ concentrates on a small span, yielding immediate near-coset structure, or via a carefully chosen quotient, the doubling constant drops polynomially. Iterative application and covering then yield the structure theorem (Moghadam, 4 Dec 2025).

4. Optimal Bounds and Limitations

The optimal exponent in Marton-type FR theorems remains unresolved. For $\mathbb{F}_2^n$, the current methods plateau at exponents between 7 and 9; achieving $C=4$ would match the best known results for integer sets via Freiman's theorem in $\mathbb{Z}$, but appears to require major new structural or entropy-combinatorial ideas (Liao, 15 Apr 2024). In entropic settings, the error bound in the structural claim is sharp up to logarithmic factors (Karam, 16 Jun 2024). For bounded-torsion groups, the dependence on $m$ in the exponent is currently $O(m^3)$ (Gowers et al., 2 Apr 2024).

5. Variations: Distributional and Symmetric-Difference Theorems

Marton-type structural theorems extend to distributional settings:

• If $A$ is a distributional $\varepsilon$-approximate group (i.e., most sums $a + b$ fall within a set $U$ of size $|A|$), then there exists a subgroup $H$ and coset $H+x$ such that $$|A \triangle (H+x)| \le 240\,\varepsilon\log(\varepsilon^{-1}|A|)|H|$$ (Karam, 16 Jun 2024).
• For small additive energy, similar symmetric-difference bounds apply.

Such results highlight that small entropy growth or approximate group properties imply strong structure, even “up to small error” in symmetric difference.

6. Connections to Classical Results and Generalizations

Classical Freiman–Ruzsa theorems establish that small doubling implies containment in a low-dimensional progression or coset. Marton-type FR theorems strengthen this by providing polynomial control in $K$. The finite-field analogues now hold with polynomial exponents, and entropic methods are unified across characteristics (Gowers et al., 2023, Moghadam, 4 Dec 2025). In prime-torsion groups, compression methods provide optimal (linear in $K$) bounds (Even-Zohar et al., 2012). For general bounded-torsion Abelian groups, multidistance entropy techniques deliver polynomial bounds (Gowers et al., 2 Apr 2024).

Open questions include extending such theorems to groups of mixed or composite torsion, removing logarithmic factors in symmetric-difference theorems, and optimizing exponents, as well as seeking analogues in nonabelian settings.

7. Summary Table of Principal Results

Group $G$	Main Theorem	Exponent	Reference
$\mathbb{F}_2^n$	Cover $A$ by $2K^C$ cosets $H$ with $|H| \le |A|$	$C=9$	(Liao, 15 Apr 2024)
$\mathbb{F}_p^n$ ($p$ odd)	Cover $A$ by $K^{O(1)}$ cosets $H$ with $|H| \le K^{O(1)}|A|$	$C=O(1)$	(Moghadam, 4 Dec 2025)
Bounded torsion $m$	Cover $A$ by $(2K)^{O(m^3)}$ cosets $H$ with $|H|\le|A|$	$O(m^3)$	(Gowers et al., 2 Apr 2024)
Distributional/entropic	$$|A \triangle (H+x)| \le O(\varepsilon\log(1/\varepsilon)|H|)$$	$O(1)$ (sharp up to log)	(Karam, 16 Jun 2024)

These theorems unify algebraic and information-theoretic approaches to small-doubling phenomena, providing a robust framework for the structure of approximate groups, coset progressions, and sumsets in finite group settings with explicit polynomial dependencies.