Polynomial Freiman-Ruzsa Conjecture Research Articles

Sumsets and entropy revisited

AbstractThe entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over .

Query complexity and the polynomial Freiman–Ruzsa conjecture

We prove a query complexity variant of the weak polynomial Freiman–Ruzsa conjecture in the following form. For any ϵ>0, a set A⊂Zd with doubling K has a subset of size at least K−4ϵ|A| with coordinate query complexity at most ϵlog2⁡|A|.We apply this structural result to give a simple proof of the “few products, many sums” phenomenon for integer sets. The resulting bounds are explicit and improve on the seminal result of Bourgain and Chang.

On iterated product sets with shifts, II

The main result of this paper is the following: for all b ∈ ℤ there exists k = k ( b ) such that max { | A ( k ) | , | ( A + u ) ( k ) | } ≥ | A | b , for any finite A ⊂ ℚ and any nonzero u ∈ ℚ . Here, | A ( k ) | denotes the k -fold product set { a 1 ⋯ a k : a 1 , … , a k ∈ A } . Furthermore, our method of proof also gives the following l ∞ sum-product estimate. For all γ > 0 there exists a constant C = C ( γ ) such that for any A ⊂ ℚ with | A A | ≤ K | A | and any c 1 , c 2 ∈ ℚ ∖ { 0 } , there are at most K C | A | γ solutions to c 1 x + c 2 y = 1 , ( x , y ) ∈ A × A . In particular, this result gives a strong bound when K = | A | 𝜖 , provided that 𝜖 > 0 is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilize a query-complexity analogue of the polynomial Freiman–Ruzsa conjecture, due to Palvolgyi and Zhelezov (2020). This new tool replaces the role of the complicated setup of Bourgain and Chang (2004), which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

From Affine to Two-Source Extractors via Approximate Duality

We establish a new connection between affine and two-source extractors by presenting black-box constructions of two-source extractors for min-entropy rate below half from any affine extractor for min-entropy rate below half. Two such constructions are presented, and one of our constructions can reach arbitrarily small min-entropy rate assuming that the affine extractor has sufficiently good parameters. The first part of our analysis shows that our constructions are two-source dispersers which are weak (but nontrivial) kinds of two-source extractors, also known as “bipartite Ramsey graphs.” To strengthen this result and obtain two-source extractors we introduce the approximate duality conjecture (ADC) and initiate its study. The ADC leads to a rather general result that can be used to convert a natural class of two-source dispersers---``low-rank dispersers''---into two-source extractors. More specifically, we first prove a special case of ADC that implies that the constructions mentioned above are two-source extractors with large (but nontrivial) constant error. In an attempt to reduce the error in our constructions we show that the polynomial Freiman--Ruzsa conjecture (PFR) in additive combinatorics implies a stronger “approximate duality” statement (and that this stronger statement also implies a weak but as-of-yet-unknown version of PFR). This stronger statement implies in turn that our constructions are two-source extractors with exponentially small error.

An Exposition of Sanders' Quasi-Polynomial Freiman-Ruzsa Theorem

The polynomial Freiman-Ruzsa conjecture is one of the most important conjec- tures in additive combinatorics. It asserts that one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying pa- rameters. This conjecture has also found several applications in theoretical computer science. Recently, Tom Sanders proved a weaker version of the conjecture, with a quasi-polynomial loss in parameters. The aim of this note is to make his proof accessible to the theoretical computer science community, and in particular to readers who are less familiar with additive combinatorics.

Equivalence of polynomial conjectures in additive combinatorics

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.

On Additive Doubling and Energy

We discuss some ideas related to the polynomial Freiman–Ruzsa conjecture. We show that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling K must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\epsilon}}$. This means that the main difficulty in proving the polynomial Freiman–Ruzsa conjecture consists of studying sets whose energy is greater than that implied by their doubling. One example is a generalized arithmetic progression of high dimension which cannot occur in the finite characteristic setting.

Some consequences of the Polynomial Freiman–Ruzsa Conjecture

Assuming the Weak Polynomial Freiman–Ruzsa Conjecture, we derive some consequences on sum-products and the growth of subsets of SL 3 ( C ) . To cite this article: M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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.