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 about the structures of ${\displaystyle A}$ and ${\displaystyle B}$? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions.

Another typical problem is to find a lower bound for ${\displaystyle |A+B|}$ in terms of ${\displaystyle |A|}$ and ${\displaystyle |B|}$. This can be viewed as an inverse problem with the given information that ${\displaystyle |A+B|}$ is sufficiently small and the structural conclusion is then of the form that either ${\displaystyle A}$ or ${\displaystyle B}$ is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different fields of mathematics, including combinatorics, ergodic theory, analysis, graph theory, group theory, and linear algebraic and polynomial methods.