Additive Number Theory - Additive Combinatorics

Additive Combinatorics

The second is principally devoted to consideration of inverse problems, often over more general groups than just the integers, that is, given some information about the sumset A+B, the aim is find information about the structure of the individual sets A and B. (A more recent name sometimes associated to this sub-division is additive combinatorics.) Unlike problems related to classical bases, as described above, this sub-area often deals with finite subsets rather than infinite ones. A typical question is what is the structure of a pair of subsets whose sumset has small cardinality (in relation to |A| and |B|). In the case of the integers, the classical Freiman's theorem provides a potent partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is simply to find a lower bound for |A+B| in terms of |A| and |B| (this can be view as an inverse problem with the given information for A+B being that |A+B| is sufficiently small and the structural conclusion then being that that either A or B is the empty set; such problems are often considered 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 draw from across the spectrum of mathematics, including combinatorics, ergodic theory, analysis, graph theory, group theory, and linear algebraic and polynomial methods.