Counting sets with small sumset and applications

We study the number of k-element sets A⊂ {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2o(k)No(1) for most N and k. As a consequence of this and a further new result concerning the number of sets A⊂ℤ/Nℤ with |A+A| ≤ c|A|2, we deduce that the random Cayley graph on ℤ/Nℤ with edge density ½ has no clique or independent set of size greater than (2+o(1)) log2N, asymptotically the same as for the Erdős-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log2N was obtained. As a second application, we show that if the elements of A ⊂ ℕ are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ℕ is equal to (2+O(1))−k/2 as k → ∞.

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