Additive Combinatorics

Many classical problems in additive number theory revolve around the study of sum sets for specific sets A, B (though one typically works with infinite sets rather than finite ones). For instance, if N∧2 ≔ {0, 1, 4, 9, 16, …} is the set of square numbers, then it is a famous theorem of Lagrange that 4N∧2 = N, i.e. every natural number is the sum of four squares; if P ≔ {2, 3, 5, 7, 11, …} is the set of prime numbers, then it is a famous theorem of Vinogradov that (2 · N + 1)\3P is finite (i.e. every sufficiently large odd number is the sum of three primes); in fact it is conjectured that this exceptional set consists only of 1, 3, and 5. The corresponding result for (2 · N)\2P remains open; the infamous Goldbach conjecture asserts that 2P contains every even integer greater than 2, but this conjecture remains far from resolution.

In this text, we shall not focus on these types of problems, which rely heavily on the specific number-theoretic structure of the sets involved. Instead, we shall focus instead on the analysis of sum sets A + B and related objects for more general sets A, B.