Sumsets

In mathematics, particularly in the area of additive combinatorics, a sumset is a set formed by the sum of elements from two or more sets.

Arithmetic combinatorics is a branch of mathematics that merges ideas from number theory and combinatorics. It focuses on the study of combinatorial problems involving integers, particularly through the lens of additive number theory and multiplicative number theory. This field investigates structures and properties of sets of integers, often using combinatorial methods to analyze problems related to arithmetic progressions, sumsets, and other additive properties.

The Davenport constant is a concept from additive number theory, particularly in the context of additive bases. It is defined in relation to a finite set of integers.

Dyson's transform, also known as the Dyson series, is a mathematical tool used primarily in quantum mechanics and quantum field theory. It provides a way to express the time evolution of a quantum state in terms of the interaction Hamiltonian when the system is subject to a time-dependent potential or interaction.

Freiman's theorem is a result in additive combinatorics that provides a structural insight into sets of integers with small sumset sizes. Specifically, it concerns the behavior of subsets of the integers that contain a large number of elements while having a relatively small sumset.

Kneser's theorem is a result in combinatorial topology and algebraic topology that deals with the intersection properties of certain families of subsets of a finite set. Specifically, it provides a bound on the size of families of subsets that can be chosen from a finite set, under the constraint that certain intersections are empty.

A **restricted sumset** is a concept used in additive combinatorics, a branch of mathematics that studies various properties of sets of numbers, particularly in relation to addition. Given two sets \(A\) and \(B\) of integers, the **sumset** \(A + B\) is defined as the set of all possible sums obtained by taking one element from \(A\) and one element from \(B\).