## Additive CombinatoricsAdditive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results. |

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Acta Arith Acta Math additive number theory algebra Amer Annals of Math B₁[g Bilu Bourgain Bull Combin Combinatorica Combinatorica 24 conjecture D.R. Heath-Brown density Discrete Comput discrete geometry Distinct distances Duke Math Elekes exponential sum finite Abelian groups Finite field finite sets Freiman's theorem GAFA Geom Gowers Graham Green Hamidoune Hungar hypergraphs inequality infinite Sidon sequence inverse theorem Israel János Bolyai Keszthely Komlós Konyagin lattice points Lecture Notes London Math Long arithmetic progressions Lower bounds Mathematics matrices Nathanson number of sums Number Theory 56 Pach Paul Erdős plane polynomial preprint prime order Probab Proc progressions in sumsets Random Structures Algorithms representation of integers Rödl Ruzsa Sárkőzy semigroups sets of integers Sidon set small sumsets Solymosi Springer-Verlag Structure Theory Studia Sci subset sums sum-free sets sum-product sums and products sums of distinct sums of sets Szemerédi Szemerédi's theorem Tardos Theory of Set Tóth triangle-free graphs Turán uncertainty principle