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Green–Tao theorem

In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for e

Restricted sumset

In additive number theory and combinatorics, a restricted sumset has the form where are finite nonempty subsets of a field F and is a polynomial over F. If is a constant non-zero function, for example

Sequences (book)

Sequences is a mathematical monograph on integer sequences. It was written by Heini Halberstam and Klaus Roth, published in 1966 by the Clarendon Press, and republished in 1983 with minor corrections

Container method

The method of (hypergraph) containers is a powerful tool that can help characterize the typical structure and/or answer extremal questions about families of discrete objects with a prescribed set of l

Schur's theorem

In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schu

Ruzsa triangle inequality

In additive combinatorics, the Ruzsa triangle inequality, also known as the Ruzsa difference triangle inequality to differentiate it from some of its variants, bounds the size of the difference of two

Gowers norm

In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure

Shapley–Folkman lemma

The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as t

Additive combinatorics

Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say

Sum-free sequence

In mathematics, a sum-free sequence is an increasing sequence of positive integers, such that no term can be represented as a sum of any subset of the preceding elements of the same sequence. This dif

Corners theorem

In arithmetic combinatorics, the corners theorem states that for every , for large enough , any set of at least points in the grid contains a corner, i.e., a triple of points of the form with . It was

Folkman's theorem

Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers are partitioned into finitel

Arithmetic combinatorics

In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis.

Sum-free set

In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A⊕A is disjoint from A. In other words, A is sum-free if the equation has no solution

Šindel sequence

In additive combinatorics, a Šindel sequence is a periodic sequence of integers with the property that its partial sums include all of the triangular numbers. For instance, the sequence that begins 1,

Szemerédi's theorem

In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with posit

Salem–Spencer set

In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free se

Approximate group

In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correc

Plünnecke–Ruzsa inequality

In additive combinatorics, the Plünnecke–Ruzsa inequality is an inequality that bounds the size of various sumsets of a set , given that there is another set so that is not much larger than . A slight

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