CS 880 : Advanced Complexity Theory 3 / 31 / 2008 Lecture 24 : Additive Combinatorics

A random A yields A+A of size closer to the high end of |A|2. Furthermore, any set A whose members are sufficiently separated has |A+A| close to this high end, and the high end is actually tight for sets A that are geometric progressions, e.g. the first n powers of two. What about the low end of |A|: what is the structure of sets A where |A+A| is close to |A|? We can see that this low-end estimate is essentially tight for sets A that are arithmetic progressions, e.g., the first n positive numbers. This brings us to another question asked by additive combinatorics: do large (high-density) sets A have to contain long arithmetic progressions? For example, a recent result shows that the set of primes — which has nontrivial density by the prime number theorem — contains arithmetic progressions of arbitrary length. Another result shows how large a set has to be to guarantee an arithmetic progression of length k. For an additive group G, let rk(G) be the maximum size of a subset A ⊆ G such that A does not contain an arithmetic progression of length k. It was conjectured that any positive-density subset of the integers from 1 to N has an arithmetic progression of length k, i.e., that