Classical and modern approaches for Plünnecke-type inequalities

Abstract

The main objective of this thesis is to present and prove Plünnecke's Inequality, a theorem that gives bounds for sumsets in commutative groups. An introduction to the theory of set addition is presented. Three different proofs of Plünnecke's Inequality are presented, two of them relying strongly on graph theory, the third being more elementary. Some other tools and techniques are introduced to obtain generalizations of Plünnecke's Inequality. The most important are the Plünnecke-Ruzsa Inequality, that gives bounds to general sum-and-difference sets in commutative groups, and some generalizations to the non-commutative case, related to Tao's Theorem. Some other generalizations involve the sum of different sets. The results are used to prove an important structural result.