Freiman's Theorem Research Articles

A strengthening of Freiman's 3k−4$3k-4$ theorem

AbstractIn its usual form, Freiman's theorem states that if and are subsets of of size with small sumset (of size close to ), then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this by allowing only a bounded number of possible summands from one of the sets. We show that if and are subsets of of size such that for any four‐element subset of the sumset has size not much more than , then already this implies that and are very close to arithmetic progressions.

A single set improvement to the 3k − 4 theorem

The 3k−4 Theorem is a classical result which asserts that if A,B⊆Z are finite, nonempty subsets with(1)|A+B|=|A|+|B|+r≤|A|+|B|+min⁡{|A|,|B|}−3−δ, where δ=1 if A and B are translates of each other, and otherwise δ=0, then there are arithmetic progressions PA and PB of common difference such that A⊆PA, B⊆PB, |B|≤|PB|+r+1 and |PA|≤|A|+r+1. It is one of the few cases in Freiman's Theorem for which exact bounds on the sizes of the progressions are known. The hypothesis (1) is best possible in the sense that there are examples of sumsets A+B having cardinality just one more than that of (1), yet A and B cannot both be contained in short length arithmetic progressions. In this paper, we show that the hypothesis (1) can be significantly weakened and still yield the same conclusion for one of the sets A and B. Specifically, if |B|≥3, s≥1 is the unique integer with(s−1)s(|B|2−1)+s−1<|A|≤s(s+1)(|B|2−1)+s, and(2)|A+B|=|A|+|B|+r<(|A|s+|B|2−1)(s+1), then we show there is an arithmetic progression PB⊆Z with B⊆PB and |PB|≤|B|+r+1. The hypothesis (2) is best possible (without additional assumptions on A) for obtaining such a conclusion.

Polylogarithmic bounds in the nilpotent Freiman theorem

AbstractWe show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup $H \subset {A^{{K^{{O_s}(1)}}}}$ modulo which ${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$ contains a nilprogression of rank at most ${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$ and size at least $\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.

An analog of Freiman's theorem in groups

An analog of Freiman's theorem in groups

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

Sets of rich lines in general position

AbstractThe Szemerédi–Trotter theorem implies that the number of lines incident to at least of points in is . J. Solymosi conjectured that if one requires the points to be in a grid formation and the lines to be in general position—no two parallel, no three meeting at a point—then one can get a much tighter bound. We prove a slight variant of his conjecture: for every there exists some such that for sufficiently large values of , every set of lines in general position, each intersecting an grid of points in at least places, has size at most . This implies a conjecture of Gy. Elekes about the existence of a uniform statistical version of Freiman's theorem for linear functions with small image sets.

Freiman's theorem in an arbitrary nilpotent group

We prove a Freiman–Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate group A in an s-step nilpotent group G is contained in a coset nilprogression of rank at most K O s ( 1 ) and cardinality at most exp ( K O s ( 1 ) ) | A | . 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.

Large values of the additive energy in and

AbstractCombining 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 ${\mathbb{Z}^d$.

Near optimal bounds in Freiman's theorem

We prove that if for a finite set A of integers we have |A+A|≤K|A|, then A is contained in a generalized arithmetic progression of dimension at most K1+C(logK)−1/2 and of size at most exp(K1+C(logK)-1/2)|A| for some absolute constant C. We also discuss a number of applications of this result.

Freiman's theorem for solvable groups

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 small doubling is is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group. As another application we establish a strengthening of the Milnor-Wolf theorem that all solvable groups of polynomial growth are virtually nilpotent, in which only one large ball needs to be of polynomial size. This result complements recent work of Breulliard-Green, Fisher-Katz-Peng, and Sanders.

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.

FREIMAN THEOREM, FOURIER TRANSFORM AND ADDITIVE STRUCTURE OF MEASURES

AbstractWe use the Freiman theorem in arithmetic combinatorics to show that if the Fourier transform of certain measures satisfies sufficiently bad estimates, then the support of the measure possesses an additive structure. The result is then discussed in light of the Falconer distance problem.

On the Freiman theorem in finite fields

For brevity, we write F2 instead of F {1,...,n} 2 . If J ⊂ I, then there is a natural embedding of F2 in F2 (defined by setting xi = 0 for i ∈ I \ J). In particular, form ≤ n, we consider F2 as a subset of F2 . LetK ≥ 1. Introduce F (K) as the minimal constant such that, for any finite set I and any nonempty A ⊂ F2 with a doubling constant σ(A) ≤ K, there is an affine subspace V ⊂ F2 containing A and of cardinality |V | ≤ F (K)|A|. The assertion claiming that F (K) is finite for anyK ≥ 1 is referred to as the Freiman theorem in F2. Ruzsa [2] proved that F (K) ≤ 2K4 for any K ≥ 1. The upper bounds for F (K) were refined in a series of papers (see [3]–[5]). Green and Tao [6] recently found the asymptotics for the logarithm of F (K) asK → ∞ by establishing the relation F (K) = 2 √ K , K ≥ 2. (1) Moreover, as was noted in [6], F (K) ≥ 22K−O(logK), K ≥ 2. (2) Here and below, log stands for the logarithm to base 2. In the present note, we refine the remainder in (1).

Appendix to ‘Roth’s theorem on progressions revisited,’ by J. Bourgain

We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-\epsilon} for all \epsilon>0.

Freiman's theorem in an arbitrary abelian group

A famous result of Freiman describes the structure of finite sets A with small doubling property. If |A + A| K|A|, then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group

On sum-product representations in $\Bbb Z_q$

The purpose of this paper is to investigate efficient representations of the residue classes modulo q , by performing sum and product set operations starting from a given subset A of \mathbb Z_q . We consider the case of very small sets A and composite q for which not much seemed known (nontrivial results were recently obtained when q is prime or when \log |A| \sim \log q ). Roughly speaking we show that all residue classes are obtained from a k -fold sum of an r -fold product set of A , where r\ll \log q and \log k \ll \log q , provided the residue sets \pi_{q'} (A) are large for all large divisors q' of q . Even in the special case of prime modulus q , some results are new, when considering large but bounded sets A . It follows for instance from our estimates that one can obtain r as small as r\sim\log q/\log|A| with similar restriction on k , something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for A=H a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on \min_{a \in \mathbb Z_q^*} \max_{x \in H} \Vert ax/q\Vert are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of x^m = a (mod q) to composite modulus q .

SETS WITH SMALL SUMSET AND RECTIFICATION

We study the extent to which sets A ⊆ Z/NZ, N prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if |A + A| ⩽ K|A| and |A| < c(K)N, then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take c ( K ) ⩾ ( 32 K ) − 12 K 2 . As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if A ⊆ F 2 n , and if |A + A| ⩽ K|A|, then A is contained in a coset of a subspace of size no more than K 2 2 2 K 2 − 2 | A | . 2000 Mathematics Subject Classification 11B75.

On a question of Erdős and Moser

For two finite sets of real numbers A and B , one says that B is sum-free with respect to A if the sum set b + b ' ∣ b , b ' ∈ B , b ≠ b ' is disjoint from A . Forty years ago, Erdőos and Moser posed the following question. Let A be a set of n real numbers. What is the size of the largest subset B of A which is sum-free with respect to A ? In this paper, we show that any set A of n real numbers contains a set B of cardinality at least g ⁡ n ln ⁡ n which is sum-free with respect to A , where g ⁡ n tends to infinity with n . This improves earlier bounds of Klarner, Choi, and Ruzsa and is the first superlogarithmic bound for this problem. Our proof combines tools from graph theory together with several fundamental results in additive number theory such as Freiman's inverse theorem, the Balog-Szemeredi theorem, and Szemeredi's result on long arithmetic progressions. In fact, in order to obtain an explicit bound on g ⁡ n , we use the recent versions of these results, obtained by Chang and by Gowers, where significant quantitative improvements have been achieved.

A multiple set version of the $3k-3$ Theorem

In 1959, Freiman demonstrated his famous 3k-4 Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a 3k-3 Theorem, proved again by Freiman. This result describes the sets of integers \mathcal{A} such that | \mathcal{A}+\mathcal{A} | \leq 3 | \mathcal{A} | -3 . In the present paper, we prove a 3k-3 -like Theorem in the context of multiple set addition and describe, for any positive integer j , the sets of integers \mathcal{A} such that the inequality |j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2 holds. Freiman's 3k-3 Theorem is the special case j=2 of our result. This result implies, for example, the best known results on a function related to the Diophantine Frobenius number. Actually, our main theorem follows from a more general result on the border of j\mathcal{A} .

The Erdős–Szeméredi problem on sum set and product set

Summary The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd˝ os-Szemeredi (E-S). (see also (El), (T), and (K-T) for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman's structure theorem) (cf. (N-T), (Na3)). We follow the reverse route and prove that if |AA| c � |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plunnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in (E-S) on the growth in k .I f