## Abstract

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 2^{o}
^{(}
^{k}
^{)}
*N*
^{o}
^{(1)} for most *N* and *k*. As a consequence of this and a further new result concerning the number of sets *A*⊂ℤ/*N*ℤ with |*A*+*A*| ≤ *c*|*A*|^{2}, we deduce that the random Cayley graph on ℤ/*N*ℤ with edge density ½ has no clique or independent set of size greater than (2+*o*(1)) log_{2}
*N*, 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 160log_{2}
*N* 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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Green, B., Morris, R. Counting sets with small sumset and applications.
*Combinatorica* **36, **129–159 (2016). https://doi.org/10.1007/s00493-015-3129-4

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### Mathematics Subject Classification (2010)

- 11P70
- 60C05
- 05A16