Binary and ternary additive problems (Q1964622)

The author shows that if all or many of the natural numbers can be written as sums of integers from given sets in many different ways, then most of those same natural numbers can be so written using small subsets of the original sets. For example, it is well-known that all, but \(\ll x/\log^{O(1)}x\), even integers \(\leq x\) can be written in \(\gg x/\log^2 x\) different ways as the sum of two primes. From this the author deduces that there is a set of \(O(\sqrt{x\log\log x})\) primes \(\leq x\) such that all but \(\ll x/\log^{O(1)}x\), even integers \(\leq x\) can be written as the sum of two primes. The proofs of the results herein are entirely combinatorial, and they come remarkably close to ``best possible'' in a rather general context. There are many further applications in the paper, and one expects that these results will be applicable in other interesting contexts.