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 2o ( 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/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)) log2 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 160log2 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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References
N. Alon, J. Balogh, R. Morris and W. Samotij: A refinement of the Cameron-Erdös Conjecture, Proc. London Math. Soc. 108 (2014), 44–72.
Y. F. Bilu, V. F. Lev and I. Z. Ruzsa: Rectification principles in additive number theory, Dedicated to the memory of Paul Erdös, Discrete Comput. Geom. 19 (1998), 343–353.
B. Bollobás and I. Leader: Compressions and isoperimetric inequalities, J. Combin. Theory Ser. A 56 (1991), 47–62.
J. Bourgain: Multilinear exponential sums in prime fields under optimal entropy condition on the sources, Geom. Funct. Anal. 18 (2009), 1477–1502.
M.-C. Chang: A polynomial bound in Freiman's theorem, Duke Math. J. 113 (2002), 399–419.
G. A. Freiman: The addition of finite sets. I (Russian), Izv. Vysš. Učebn. Zaved. Matematika 13 (1959), 202–213.
M. Garaev: A quantified version of Bourgain's sum-product esimate in Fp for subsets of incomparable sizes, Electronic J. Combinatorics 15 (2008).
S. W. Graham and C. J. Ringrose: Lower bounds for least quadratic non-residues, in: Analytic Number Theory (Allerton Park 1989), Progress in Mathematics 85, 269–309, Birkhäuser (Basel) 1990.
B. J. Green: Edinburgh lecture notes on Freiman’s theorem, in preparation for Online J. Analytic Combinatorics.
B. J. Green: Counting sets with small sumset, and the clique number of random Cayley graphs, Combinatorica 25 (2005), 307–326.
B. J. Green: A Szemerédi-type regularity lemma in abelian groups, Geom. Funct. Anal. 15 (2005), 340–376.
B. J. Green and I. Z. Ruzsa: Counting sumsets and sum-free sets modulo a prime, Studia Sci. Math. Hungarica 41 (2004), 285–293.
B. J. Green and I. Z. Ruzsa: Sets with small sumset and rectification, Bull. London Math. Soc. 38 (2006), 43–52.
B. J. Green and T. C. Tao: An arithmetic regularity lemma, associated counting lemma, and applications, in: An irregular mind — Szemerédi is 70, Bolyai Soc. Math. Stud. 21, 261–334, János Bolyai Math. Soc., Budapest, 2010.
M. Krivelevich, B. Sudakov, V. H. Vu and N. C. Wormald: Random regular graphs of high degree, Random Structures Algorithms 18 (2001), 346–363.
J. M. Pollard: A generalization of the theorem of Cauchy and Davenport, J. London Math. Soc. 2 (1974), 460–462.
I. Z. Ruzsa: Arithmetical progressions and the number of sums, Period. Math. Hungar. 25 (1992), 105–111.
T. Sanders: The structure theory of set addition revisited, Bull. Amer. Math. Soc. 50 (2013), 93–127.
T. Schoen: The cardinality of restricted sumsets, J. Number Theory 96 (2002), 48–54.
T. C. Tao and V. H. Vu: Additive combinatorics, Cambridge University Press, 2006.
D.-L Wang and P. Wang: Discrete isoperimetric problems, SIAM J. Appl. Math. 32 (1977), 860–870.
