Additive combinatorics

Additive combinatorics (from the English addition - addition) is an interdisciplinary ^[1] area of mathematics that studies the interdependence of various quantitative interpretations of the concept of structured subsets of a group (usually finite), as well as similar properties of derivatives of many structures used in these interpretations. In addition, additive combinatorics studies the structuredness in various senses of some specific sets or classes of sets (for example, subsets of primes or multiplicative subgroups ).

Thus, the main object of study is finite sets, as abstract as possible, sometimes limited only in size, which makes this science look like combinatorics . However, unlike combinatorics as such, where the elements of sets are identified only by their difference from each other and belonging to one or another of the considered sets, in additive combinatorics each element of the set has a clear meaning, and additional relationships appear between the elements arising from their values and properties of the operation of the group (and, possibly, specific laws characteristic of a given group).

Additive combinatorics is closely related to arithmetic combinatorics , where the subject of study is the ratio of a subset of a field (and not just a group, as here) to two operations at once - addition and multiplication.

Prerequisites

Additive Number Theory

The issues of representing numbers through sums of elements from a small number of given sets have been considered by mathematicians since ancient times. Classical examples are the problems of the representability of any number by the sum of four squares ( the Lagrange theorem on the sum of four squares ) or any even number that is more than three by the sum of two simple ( Goldbach problem ). If denoted by ${\mathbb {Q} }=\left\lbrace {0}\right\rbrace \cup \left\lbrace {n^{2}:n\in {\mathbb {N} }}\right\rbrace$ ${\ displaystyle {\ mathbb {Q}} = \ left \ lbrace {0} \ right \ rbrace \ cup \ left \ lbrace {n ^ {2}: n \ in {\ mathbb {N}}} \ right \ rbrace }$ ${\mathbb {Q} }=\left\lbrace {0}\right\rbrace \cup \left\lbrace {n^{2}:n\in {\mathbb {N} }}\right\rbrace$ set of all squares of non-negative numbers, then in terms of additive combinatorics (see the notation section below), the Lagrange theorem can be written as

{\mathbb {Q} }+{\mathbb {Q} }+{\mathbb {Q} }+{\mathbb {Q} }={\mathbb {N} }

{\ displaystyle {\ mathbb {Q}} + {\ mathbb {Q}} + {\ mathbb {Q}} + {\ mathbb {Q}} = {\ mathbb {N}}}

{\mathbb {Q} }+{\mathbb {Q} }+{\mathbb {Q} }+{\mathbb {Q} }={\mathbb {N} }

In the course of solving similar problems, other similar ones arose, with different sets, but similar formulations. All this has formed a field of mathematics called .

However, additive combinatorics should not be taken as a generalization or development of additive number theory - although the problems of the latter can be conveniently written in terms of additive combinatorics, but its general methods, as a rule, are not applicable to them. Number theory always considers sets of a special kind - prime numbers, squares, other degrees, numbers with a small number of divisors, etc. Additive combinatorics try to understand the structure of addition as such, to derive the most general results.

Nevertheless, the first results, which can be related in spirit to additive-combinatorial ones, arose at the beginning of the 20th century precisely as part of number theory (also called combinatorial number theory). ^[1] So, for example, is the Schnirelman method for solving the Goldbach problem (which was also used by Linnik for the Waring problem ) - it is based on the Schnirelman theorem on the set of sums of numbers from two arbitrary sets, of which only their density is known ^[2] (follows note that in this case, the very specific definition of density according to Schnirelman, used in this theorem, did not take root in additive combinatorics as an object for study).

Ramsey Theory

The Ramsey theory that arose in the first half of the 20th century also analyzed various ideas about structuredness. The statements of the Ramsey theory relate to the presence of at least a small “island” of structurality in rather complex (in terms of the number of elementary parts) objects. ^[3]

However, Ramsey’s theory does not consider subsets, but partitions of a given set (for example, the set of edges of a graph, natural numbers or part of a Boolean of a finite set), and the result on the structure means the structure of one of the elements of the partition, and, as a rule, it is not clear that which one. Consequently, nothing can be said about any particular subset.

A good example is - it says that for any partition of natural numbers into a finite number of classes, at least one of the classes will have an arithmetic progression (of any given length). Moreover, it is obvious that in any partition there is a class whose number density is greater than in others, and intuitively it seems that progression should be in this set - after all, there are most of all elements, that is, most of all possibilities. It is also obvious that the largest set will have a positive (nonzero) density. However, to prove that absolutely any infinite set of positive integers of positive density contains an arithmetic progression cannot be obtained simply through the van der Waerden theorem - according to it, the progression can be in any class, even the smallest. To prove the presence of progression in any dense set, one has to use much more complex means - the solution to this problem is called the Szemeredi theorem , which is considered to be the classical additive-combinatorial result.

Trigonometric Amounts

A flexible tool for evaluating the ordering of a set, which played a significant role in additive number theory, are trigonometric sums (sums of the roots of unity corresponding to the numbers of the set). They became a tool and object for study in additive combinatorics, since they allow a fairly general application.

Even Gauss, the first to study trigonometric sums, discovered through them the relationship of the distribution of all possible differences of numbers from a given set and the set itself. Examining quadratic residues, Gauss considered the sum

S(a)=\sum \limits _{x=0}^{p-1}{e^{2\pi {\frac {ax^{2}}{p}}i}}

{\ displaystyle S (a) = \ sum \ limits _ {x = 0} ^ {p-1} {e ^ {2 \ pi {\ frac {ax ^ {2}} {p}} i}}}

and derived an estimate for the square of its module:

|S(a)|^{2}=\sum \limits _{x,y}{e^{2\pi {\frac {a(x^{2}-y^{2})}{p}}i}}

{\ displaystyle | S (a) | ^ {2} = \ sum \ limits _ {x, y} {e ^ {2 \ pi {\ frac {a (x ^ {2} -y ^ {2})} {p}} i}}}

An estimate for this sum was obtained purely combinatorially from the properties of the expression $x^{2}-y^{2}$ ${\ displaystyle x ^ {2} -y ^ {2}}$ when replacing a variable $x=y+h$ ${\ displaystyle x = y + h}$ . ^[4] Thus, the multiset of differences gave information about the structure of the set of quadratic residues themselves. In additive combinatorics, a similar approach is applied in principle, when there is already a set rather than a multiset of differences (or sums, products, etc.) of elements from a given set gives information about the structure of this set. Such a transition - from a multiset to a set - is associated with a transition from the number of solutions of a given equation (for example, $a_{1}+b_{1}=a_{2}+b_{2},a_{i}\in A,b_{i}\in B$ ${\ displaystyle a_ {1} + b_ {1} = a_ {2} + b_ {2}, a_ {i} \ in A, b_ {i} \ in B}$ ) to the representation of numbers in one form or another (for example, in the form $a+b,a\in A,b\in B$ ${\ displaystyle a + b, a \ in A, b \ in B}$ ), which in principle is characteristic of the method of trigonometric sums.

Freiman's theorem

A separate foundation for the development of a new science of element-wise sums of sets (sets of sums $A+B=\left\lbrace {a+b:a\in A,b\in B}\right\rbrace$ ${\ displaystyle A + B = \ left \ lbrace {a + b: a \ in A, b \ in B} \ right \ rbrace}$ ) became the theorem proved by Gregory Freiman on the structure of sets with small doubling (that is, sets whose set of sums of two elements is small in size, or more simply, the sums of elements of which often coincide). ^[five]

Questions about the sums of elements from a given set were also considered by Erdös and Semeredi without introducing special symbols to denote the summation of sets. ^[6]

Study Subject

Many amounts

An important concept of additive combinatorics is the set of sums.

A+B=\left\lbrace {a+b:a\in A,b\in B}\right\rbrace

{\ displaystyle A + B = \ left \ lbrace {a + b: a \ in A, b \ in B} \ right \ rbrace}

for finite sets $A,B\in \mathrm {G}$ ${\ displaystyle A, B \ in \ mathrm {G}}$ where $(\mathrm {G} ,+)$ ${\ displaystyle (\ mathrm {G}, +)}$ - Group. The size of such a set is determined by the structure $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ regarding operation $+$ ${\ displaystyle +}$ . If a $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ - arithmetic progressions then $|A+B|$ ${\ displaystyle | A + B |}$ few. And if, for example, $A,B\in {\mathbb {Z} }_{n}$ ${\ displaystyle A, B \ in {\ mathbb {Z}} _ {n}}$ randomly selected according to the Bernoulli scheme then $|A+B|$ ${\ displaystyle | A + B |}$ very large. The intermediate cases between the two indicated are precisely additive combinatorial interest.

In addition, the set structure itself is interesting. $A+B$ ${\ displaystyle A + B}$ . In particular, as of 2018, there are no general criteria (other than direct enumeration) to determine whether a given set is represented in the form $A+A$ ${\ displaystyle A + A}$ .

Associated Characteristics of Sets

The table below summarizes the theorems and inequalities of additive combinatorics that relate the various characteristics of subsets of groups. The statement indicated in the cell links the characteristics indicated in its row and column. Only some of the most famous of these theorems are presented.

For brevity, the headings use the following abbreviations:

Set density $A\subset {\mathbb {G} }$ ${\ displaystyle A \ subset {\ mathbb {G}}}$ for the final group ${\mathbb {G} }$ ${\ displaystyle {\ mathbb {G}}}$ Is a quantity ${\frac {|A|}{|{\mathbb {G} }|}}$ ${\ displaystyle {\ frac {| A |} {| {\ mathbb {G}} |}}}$ or the law of its asymptotic growth with increasing size ${\mathbb {G} }$ ${\ displaystyle {\ mathbb {G}}}$ , but for infinite ${\mathbb {G} }$ ${\ displaystyle {\ mathbb {G}}}$ - asymptotic density (or distribution law) $A$ ${\ displaystyle A}$ at ${\mathbb {G} }$ ${\ displaystyle {\ mathbb {G}}}$ ,
OAP (from "generalized arithmetic progression") - the presence in a set of large arithmetic progressions , non-trivial generalized arithmetic progressions or their large parts, and also, on the contrary, the ability to cover the set (or most of it) not with a big arithmetic progression;
$A_{1}+\dots +A_{k}$ ${\ displaystyle A_ {1} + \ dots + A_ {k}}$ - the size and structure of the set of sums (as well as differences and combinations of sums and differences) - in particular, the ability to represent any element of the group as the sum of a given number of elements of this set;
$X=A_{1}+\dots +A_{k}$ ${\ displaystyle X = A_ {1} + \ dots + A_ {k}}$ - the representability of a set or a large part thereof as a set of sums (as well as differences and combinations of sums and differences) of numbers from a small number of sets, i.e., the solvability of the equation of sets $X=A_{1}+\dots +A_{k}$ ${\ displaystyle X = A_ {1} + \ dots + A_ {k}}$ for a given $X$ ${\ displaystyle X}$ perhaps even under certain conditions on $A_{1},\dots ,A_{k}$ ${\ displaystyle A_ {1}, \ dots, A_ {k}}$ (eg, $A_{1}=\dots =A_{k}$ ${\ displaystyle A_ {1} = \ dots = A_ {k}}$ ), where $+$ ${\ displaystyle +}$ means the amount of Minkowski
Fourier coefficients are meant for the characteristic function of the set and the functions defined through it, as well as their statistics - various kinds of norms , the number of elements with a large value and the structure of their set;
CRO (from "the number of solutions of the equation") - the number of solutions of various linear equations (in particular, additive energy ) or systems of equations in which variables take values from given sets, as well as the ratio of the number of their solutions

Also in cells several specific designations are used:

${\widehat {A}}$ ${\ displaystyle {\ widehat {A}}}$ Is the Fourier coefficient of the characteristic function of the set $A$ ${\ displaystyle A}$ ;
$E(A,B)$ ${\ displaystyle E (A, B)}$ - additive energy
$\rho _{A}(x)=\mathbf {1} _{A}(x)-{\frac {|A|}{|{\mathbb {G} }|}}$ ${\ displaystyle \ rho _ {A} (x) = \ mathbf {1} _ {A} (x) - {\ frac {| A |} {| {\ mathbb {G}} |}}}$ - such a function is called the balance function of the set $A\in {\mathbb {G} }$ ${\ displaystyle A \ in {\ mathbb {G}}}$ , insofar as $\sum \limits _{x\in {\mathbb {G} }}{\rho _{A}(x)}=0$ ${\ displaystyle \ sum \ limits _ {x \ in {\ mathbb {G}}} {\ rho _ {A} (x)} = 0}$ .


	Density	OAP	$A_{1}+\dots +A_{k}$ ${\ displaystyle A_ {1} + \ dots + A_ {k}}$	$X=A_{1}+\dots +A_{k}$ ${\ displaystyle X = A_ {1} + \ dots + A_ {k}}$	Fourier coefficients	CHRU
Density
OAP	Szemeredi theorem
$A_{1}+\dots +A_{k}$ ${\ displaystyle A_ {1} + \ dots + A_ {k}}$	Schnirelman Inequality ,
$X=A_{1}+\dots +A_{k}$ ${\ displaystyle X = A_ {1} + \ dots + A_ {k}}$		$A+B$ ${\ displaystyle A + B}$ at large $\|A\|$ ${\ displaystyle \| A \|}$ and $\|B\|$ ${\ displaystyle \| B \|}$ contain a long a. p. ^[7] ^[8]	Plynneke Inequality - Rouge
Fourier coefficients	The uncertainty principle ^[9]	If a $A\subset {\mathbb {Z} }_{N}$ ${\ displaystyle A \ subset {\ mathbb {Z}} _ {N}}$ , $\|\|{\widehat {\rho _{A}}}\|\|_{\infty }$ ${\ displaystyle \|\| {\ widehat {\ rho _ {A}}} \|\| _ {\ infty}}$ little then $A$ ${\ displaystyle A}$ contains a. n. length 3 ^[10]	If a $\|\|{\widehat {A}}\|\|_{\infty }$ ${\ displaystyle \|\| {\ widehat {A}} \|\| _ {\ infty}}$ and $\|\|{\widehat {B}}\|\|_{\infty }$ ${\ displaystyle \|\| {\ widehat {B}} \|\| _ {\ infty}}$ small then $\|A+B\|$ ${\ displaystyle \| A + B \|}$ great
CHRU	Roth's theorem	If a $A$ ${\ displaystyle A}$ - but. n. then $E(A,A)\geq \Omega (\|A\|^{3})$ ${\ displaystyle E (A, A) \ geq \ Omega (\| A \| ^ {3})}$	$\|A\|^{2}\|B\|^{2}\geq \|A+B\|E(A,B)$ ${\ displaystyle \| A \| ^ {2} \| B \| ^ {2} \ geq \| A + B \| E (A, B)}$ ^[eleven]	From the ratios of additive energies, conclusions can be drawn about the structure of the set ^[12]	If a $A,B\in {\mathbb {Z} }_{N}$ ${\ displaystyle A, B \ in {\ mathbb {Z}} _ {N}}$ then $E(A,B)=\sum \limits _{r=0}^{N-1}{\|{\widehat {A}}(r)\|^{2}\|{\widehat {B}}(r)\|^{2}}$ ${\ displaystyle E (A, B) = \ sum \ limits _ {r = 0} ^ {N-1} {\| {\ widehat {A}} (r) \| ^ {2} \| {\ widehat {B} } (r) \| ^ {2}}}$

Some concepts used

is a generalization of the concept of Fourier coefficients, coined by William Gowers in the course of the proof of Szemerédi's theorem.

Freiman isomorphism is a mapping from a subset of one group to a subset of another, preserving the relation of equality of the sums of a given number of elements of the set.

Final set $A=\left\lbrace {a_{1}\leq \dots \leq a_{n}}\right\rbrace$ ${\ displaystyle A = \ left \ lbrace {a_ {1} \ leq \ dots \ leq a_ {n}} \ right \ rbrace}$ real numbers is called a strictly convex set if $a_{i+1}-a_{i}<a_{i+2}-a_{i+1}$ ${\ displaystyle a_ {i + 1} -a_ {i} <a_ {i + 2} -a_ {i + 1}}$ for $i=1,\dots ,n-2$ ${\ displaystyle i = 1, \ dots, n-2}$ that is, if $A$ ${\ displaystyle A}$ is a cut image $\left\lbrace {1,\dots ,n}\right\rbrace$ ${\ displaystyle \ left \ lbrace {1, \ dots, n} \ right \ rbrace}$ for some strictly convex function . ^{[13] The} properties of such sets are widely studied in additive combinatorics. ^[14] ^[15] ^[16] ^[17] This concept should not be confused with a convex set in the usual sense .

The Bohr set is a small doubling structure, sometimes used in evidence as a weakened analogue of the concept of subspace. ^[18] . It is defined as the set of field elements on which all additive characters from a given family are of little importance. Bohr sets contain large generalized arithmetic progressions, so the presence of a set of progressions is sometimes proved through the presence of the necessary Bohr set in it.

function $f$ ${\ displaystyle f}$ such that the norm $||f(x+t)-f(x)||_{p}$ ${\ displaystyle || f (x + t) -f (x) || _ {p}}$ small enough for some $p$ ${\ displaystyle p}$ and for everyone $t\in T$ ${\ displaystyle t \ in T}$ where $T$ ${\ displaystyle T}$ - some set (for example, a set of Bohr). On such sets one of the proofs of Roth's theorem is constructed. ^[nineteen]

The set of large trigonometric sums (sometimes also called the spectrum ) of the set $A\subset {\mathbb {Z} }_{p}$ ${\ displaystyle A \ subset {\ mathbb {Z}} _ {p}}$ Is a lot of deductions $r$ ${\ displaystyle r}$ for which the amount $\sum \limits _{a\in A}{e_{p}(ar)}$ ${\ displaystyle \ sum \ limits _ {a \ in A} {e_ {p} (ar)}}$ (Fourier coefficient) has a large modulus . ^[20]

A dissociative set is a set in which the sums of elements from two different subsets are always different, that is, a sum of the form $\sum \limits _{a\in A}{\varepsilon _{a}a},\forall a\in A:\varepsilon _{a}\in \left\lbrace {-1,0,1}\right\rbrace$ ${\ displaystyle \ sum \ limits _ {a \ in A} {\ varepsilon _ {a} a}, \ forall a \ in A: \ varepsilon _ {a} \ in \ left \ lbrace {-1,0,1 } \ right \ rbrace}$ never equal to zero. ^[20]

Learning Methods

Elementary Methods

Of course, despite the existence of powerful and complex methods for studying additive combinatorics theorems, some methods and tasks lend themselves to an elementary description. From Cauchy's inequality $\left({\sum \limits _{i=1}^{n}{a_{i}}}\right)^{2}\leq n\sum \limits _{i=1}^{n}{{a_{i}}^{2}}$ ${\ displaystyle \ left ({\ sum \ limits _ {i = 1} ^ {n} {a_ {i}}} \ right) ^ {2} \ leq n \ sum \ limits _ {i = 1} ^ { n} {{a_ {i}} ^ {2}}}$ properties of additive energy are deduced that are applied almost universally. In general, the elementary approach often analyzes the number of solutions of certain equations. The is also often used — for example, when decomposing the number of solutions of an equation by the sum of the number of solutions for a given value of a particular variable. ^[21]

By elementary methods, one can prove Roth's theorem ^[22] and the Cauchy – Davenport theorem ^[23] .

However, many evidence obtained by other methods have no elementary analogues.

Combinatorial methods

One of the most significant combinatorial proofs of additive combinatorics is the first evidence that appeared to be the Cemeredi theorem ^[24] - in it, Cemeredi formulated and used the so-called regularity lemma concerning pure graph theory . Graphical analogies are also used to prove special versions of the Plunknecke-Rouge inequality or estimates like Balog-Szemeredi-Gowers ^[25] .

Algebraic methods

Algebraic methods in additive combinatorics use the properties of polynomials that are built on the basis of various sets. The degrees of such polynomials, as a rule, depend on the sizes of the sets under study, and the set of roots of the polynomials can give some kind of information about the sums, intersections, etc. of the original sets.

An example of a tool for applying this method is the combinatorial zeros theorem . Using it, the Cauchy-Davenport theorem and some of its generalizations can be proved. ^[23]

Other applications of the algebraic method can be found in the proofs of Roth's theorem for some groups of a special kind ^[26] ^[27] ^[28] or in the estimate of the size of intersections of shifts of multiplicative subgroups of fields ${\mathbb {F} }_{p}$ ${\ displaystyle {\ mathbb {F}} _ {p}}$ and the proof of the Waring problem for a simple field (for this, in particular, the Stepanov method is used ). ^[29]

Analytical Methods

The main analytical tool in additive combinatorics is the Fourier coefficients . This is due to the close relationship between the operation of taking the Fourier coefficient and the operation of convolution of functions . The Fourier coefficients were applied as early as the first proof of Roth's theorem. ^[30] Their generalization is Gowers norms, the science of which is also called higher-order Fourier analysis. ^[20]

By the example of Fourier coefficients (in particular, their application to the proof of Roth's theorem), the so-called iterative argument is best illustrated when the consideration of an arbitrary set is divided into two cases - when the set does not have large (relative to the size of the set) Fourier coefficients and when it is. In the first case, you can directly use the properties of the Fourier coefficients, and in the second, you can find the substructure of a set with a higher density in an infinite arithmetic progression, and with so much that the number of such possible steps to reach the limit density will be limited by a value depending on the total density of the initial set. This reveals the idea of separation into the case of a pseudo-random set and a set with a certain global structure. This idea was reflected in other methods. ^[1] ^[10]

Another analytical approach is to study the almost periodicity of functions associated with the characteristic functions of the sets under study. ^[31]

Ergodic methods

The ergodic approach involves applying results from the theory of dynamical systems to problems of additive combinatorics. This approach was first applied by Hillel Fürstenberg to the Cemeredi theorem ^[32] , but it soon turned out that it can be generalized to a much wider range of problems.

The theory of dynamical systems often allows one to prove results that are not accessible for reformulation by other methods, but is not able to give any quantitative estimates (for example, estimates for the density of a set in Cemeredi's theorem). ^[33]

Other methods

To the science in question have applications and some other areas:

Szemeredi – Trotter theorem from combinatorial geometry (especially in questions concerning convex sets) ^[16] ^[13] ^[17] ;
geometry of numbers ^[34] ;
^[35] .
theory of eigenvalues (can be used to derive inequalities between the number of solutions of some systems of linear equations; ^[36] );

Some researchers

Paul Erdös
Grigory Freiman
Endre semeredi
William Gowers
Ben green
Terence Tao
Jean burgain
Sergey Konyagin
Ilya Shkredov

Literature

Graham, Ronald. The beginning of the Ramsey theory. - M .: Mir, 1984. - ISBN 5-09-002630-0 .

M.Z. Garaev. Sums and products of sets and estimates of rational trigonometric sums in fields of simple order (Russian) // Uspekhi Matematicheskikh Nauk. - 2010. - Issue. 4 . - S. 5–66 .

I. D. Shkredov. Semeredi theorem and problems on arithmetic progressions (rus.) // Uspekhi Matematicheskikh Nauk. - 2006. - Vol. 6 . - S. 111-179 .

I. D. Shkredov. Fourier analysis in combinatorial number theory (rus.) // Successes in Mathematical Sciences. - 2010. - Issue. 3 . - S. 127-184 .

Terence Tao , Van H. Vu. Additive combinatorics . - Cambridge University Press. - 2006. - ISBN 0-511-24530-0 .

Imre Ruzsa. Sumsets and structure .

Freiman, Grigory Abelevich . The beginning of the structural theory of sets (Russian) . - Printing house Tatpolygraph. - 1966.

I. M. Gelfand , Yu. V. Linnik . Elementary methods in analytical number theory (rus.) . - M .: Fizmatgiz. - 1962.

I.M. Vinogradov . The method of trigonometric sums in number theory (Russian) . - Publishing house "Science". - 1971.

Notes

↑ ¹ ² ³ Post-science, I. D. Shkredov, “Additive combinatorics”
↑ Gelfand, 1962 , p. 9-43.
↑ Graham, 1984 .
↑ Vinogradov, 1971 , p. 5-6.
↑ Freiman, 1966 .
↑ Erdős, Paul & Szemerédi, Endre (1983), "On sums and products of integers" , Studies in Pure Mathematics. To the memory of Paul Turán , Basel: Birkhäuser Verlag, p. 213–218, ISBN 978-3-7643-1288-6 , doi : 10.1007 / 978-3-0348-5438-2_19 .
↑ Ernie Croot, Izabella Laba, Olof Sisask, "Arithmetic Progressions in Sumsets and L ^ p-Almost-Periodicity"
↑ Tom Sanders, "Additive structures in sumsets"
↑ Terence Tao, "An uncertainty principle for cyclic groups of prime order"
↑ ¹ ² http://www.mathnet.ru/php/seminars.phtml?presentid=3055 Meetings of the Moscow Mathematical Society, March 1, 2011, I. D. Shkredov, Methods of additive combinatorics
↑ Garaev, 2010 , p. 25.
↑ All- Institute Seminar "Mathematics and its Applications" of the Mathematical Institute named after V. A. Steklova RAS, 09/18/14, I. D. Shkredov, "Structural theorems in additive combinatorics"
↑ ¹ ² A. Iosevich, S. Konyagin, M. Rudnev, and V. Ten, “On combinatorial complexity of convex sequences”, July 19, 2004
↑ MZ Garaev, “On Lower Bounds for the L1-Norm of Exponential Sums”, Mathematical Notes, November 2000, Volume 68, Issue 5-6, pp 713-720
↑ Imre Z. Ruzsa, Dmitrii Zhelezov, “Convex sequences may have thin additive bases”, preprint
↑ ¹ ² Tomasz Schoen, Ilya Shkredov, “On Sumsets of Convex Sets”
↑ ¹ ² Elekes, Nathanson, Ruzsa, “Convexity and sumsets”
↑ Ben Green, “Finite field models in additive combinatorics”
↑ Tom Sanders, “On Roth's theorem on progressions”, preprint
↑ ¹ ² ³ Shkredov, 2010 .
↑ Garaev, 2010 .
↑ Graham, 1984 , p. 20.
↑ ¹ ² Mathematical Laboratory named after P. L. Chebyshev, Lecture Course “Additive Combinatorics”, Lecture 1
↑ Szemerédi, Endre (1975), " On sets of integers containing no k elements in arithmetic progression ", Acta Arithmetica T. 27: 199–245, Zbl 0303.10056, MR : 0369312 , < http://matwbn.icm.edu. pl / ksiazki / aa / aa27 / aa27132.pdf > .
↑ Garaev, 2010 , p. 18-25.
↑ E. Croot, V. Lev, and PP Pach, Progression-free sets in ${\mathbb {Z} }_{n}^{4}$ ${\ displaystyle {\ mathbb {Z}} _ {n} ^ {4}}$ are exponentially small (2016). arXiv preprint 1605.01506.
↑ Proof of Roth's theorem for small torsion groups at arxiv.org
↑ Statement of the proof of Roth's theorem for ${\mathbb {F} }_{4}^{n}$ ${\ displaystyle {\ mathbb {F}} _ {4} ^ {n}}$ in Russian
↑ I. V. Vyugin, I. D. Shkredov, “On additive shifts of multiplicative subgroups”, Mat. Sat, 2012, Volume 203, Number 6, Pages 81-100
↑ Shkredov, 2006 , p. 119-124.
↑ I. D. Shkredova, review lecture "Analytical methods in additive combinatorics", lecture hall of mathematical laboratory named after Chebyshev
↑ Yufei Zhao, “Szemer´edi's Theorem via Ergodic Theory”
↑ Post-science, I. D. Shkredov, “Combinatorial Ergodic Theory”
↑ Imre Ruzsa, “Additive combinatorics and geometry of numbers”
↑ JA Dias da Silva, YO Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994) 140-146
↑ I. D. Shkredov, "Several new results on higher energies"

[Shkredov,Postnauka,AC-1] ¹ ² ³ Post-science, I. D. Shkredov, “Additive combinatorics”

[_d93e23b89c89fda5-2] Gelfand, 1962 , p. 9-43.

[_f6a9a599e819a10c-3] Graham, 1984 .

[_2364855db7a04fd9-4] Vinogradov, 1971 , p. 5-6.

[_a64063c7b6fd258d-5] Freiman, 1966 .

[6] Erdős, Paul & Szemerédi, Endre (1983), "On sums and products of integers" , Studies in Pure Mathematics. To the memory of Paul Turán , Basel: Birkhäuser Verlag, p. 213–218, ISBN 978-3-7643-1288-6 , doi : 10.1007 / 978-3-0348-5438-2_19 .

[7] Ernie Croot, Izabella Laba, Olof Sisask, "Arithmetic Progressions in Sumsets and L ^ p-Almost-Periodicity"

[8] Tom Sanders, "Additive structures in sumsets"

[9] Terence Tao, "An uncertainty principle for cyclic groups of prime order"

[Shkredov,methods-10] ¹ ² http://www.mathnet.ru/php/seminars.phtml?presentid=3055 Meetings of the Moscow Mathematical Society, March 1, 2011, I. D. Shkredov, Methods of additive combinatorics

[_b9ec7db827608142-11] Garaev, 2010 , p. 25.

[12] All- Institute Seminar "Mathematics and its Applications" of the Mathematical Institute named after V. A. Steklova RAS, 09/18/14, I. D. Shkredov, "Structural theorems in additive combinatorics"

[IKRT-13] ¹ ² A. Iosevich, S. Konyagin, M. Rudnev, and V. Ten, “On combinatorial complexity of convex sequences”, July 19, 2004

[14] MZ Garaev, “On Lower Bounds for the L1-Norm of Exponential Sums”, Mathematical Notes, November 2000, Volume 68, Issue 5-6, pp 713-720

[15] Imre Z. Ruzsa, Dmitrii Zhelezov, “Convex sequences may have thin additive bases”, preprint

[Schoen_Shkredov_convex-16] ¹ ² Tomasz Schoen, Ilya Shkredov, “On Sumsets of Convex Sets”

[ENR,_1999-17] ¹ ² Elekes, Nathanson, Ruzsa, “Convexity and sumsets”

[18] Ben Green, “Finite field models in additive combinatorics”

[19] Tom Sanders, “On Roth's theorem on progressions”, preprint

[_d040d16649d802d0-20] ¹ ² ³ Shkredov, 2010 .

[_94349a0138fe46e5-21] Garaev, 2010 .

[_538588a828005712-22] Graham, 1984 , p. 20.

[Petrov1-23] ¹ ² Mathematical Laboratory named after P. L. Chebyshev, Lecture Course “Additive Combinatorics”, Lecture 1

[24] Szemerédi, Endre (1975), " On sets of integers containing no k elements in arithmetic progression ", Acta Arithmetica T. 27: 199–245, Zbl 0303.10056, MR : 0369312 , < http://matwbn.icm.edu. pl / ksiazki / aa / aa27 / aa27132.pdf > .

[_e827e0286f3a9843-25] Garaev, 2010 , p. 18-25.

[26] E. Croot, V. Lev, and PP Pach, Progression-free sets in ${\mathbb {Z} }_{n}^{4}$ ${\ displaystyle {\ mathbb {Z}} _ {n} ^ {4}}$ are exponentially small (2016). arXiv preprint 1605.01506.

[27] Proof of Roth's theorem for small torsion groups at arxiv.org

[28] Statement of the proof of Roth's theorem for ${\mathbb {F} }_{4}^{n}$ ${\ displaystyle {\ mathbb {F}} _ {4} ^ {n}}$ in Russian

[29] I. V. Vyugin, I. D. Shkredov, “On additive shifts of multiplicative subgroups”, Mat. Sat, 2012, Volume 203, Number 6, Pages 81-100

[_7640b6d0ccbaebb1-30] Shkredov, 2006 , p. 119-124.

[31] I. D. Shkredova, review lecture "Analytical methods in additive combinatorics", lecture hall of mathematical laboratory named after Chebyshev

[32] Yufei Zhao, “Szemer´edi's Theorem via Ergodic Theory”

[33] Post-science, I. D. Shkredov, “Combinatorial Ergodic Theory”

[34] Imre Ruzsa, “Additive combinatorics and geometry of numbers”

[35] JA Dias da Silva, YO Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994) 140-146

[36] I. D. Shkredov, "Several new results on higher energies"