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Sums and difference of finite sets. (English) Zbl 1226.11019

The problem of estimating cardinality of sumsets is one of the interesting and difficult topics of the additive number theory. In this paper, the authors deal with a particular case of this problem. Let \(A\) and \(B\) be two finite subsets in a given abelian group, by using Plünnecke inequalities as the most important tool, they consider the question of comparing the size of \(A-B\) with that of \(A+B\) and the one of estimating the ratio \(|X-B|/|X|\) when \(X\) runs over all the non-empty subsets of \(A\), where \(A\) and \(B\) satisfy the small sumset condition \(|A+B|\leq k|A|\) .

MSC:

11B13 Additive bases, including sumsets
11B75 Other combinatorial number theory
11P70 Inverse problems of additive number theory, including sumsets
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References:

[1] G.A. Freiman, W.P. Pigarev, The relation between the invariants \(R\) and \(T\), (Russian), Kalinin. Gos. Univ. Moscow (1973), 172–174. · Zbl 0321.10048
[2] F. Hennecart, G. Robert, A. Yudin, On the number of sums and differences. In Structure theory of set addition . Astérisque No. 258 (1999), 173–178. · Zbl 0969.11011
[3] I.Z. Ruzsa, On the cardinality of \(A+A\) and \(A-A\). In Coll. Math. Soc. Bolyai 18 , Combinatorics (Keszthely 1976), Akadémiai Kiadó (Budapest 1979), 933–938. · Zbl 0383.04003
[4] I.Z. Ruzsa, An application of graph theory to additive number theory. Scientia 3 (1989), 97–109. · Zbl 1103.05316
[5] I.Z. Ruzsa, On the number of sums and differences. Acta Math. Hungar. 59 (1992), 439–447. · Zbl 0773.11010 · doi:10.1007/BF00050906
[6] I.Z. Ruzsa, Sums of finite sets. In Number theory (New York, 1991–1995), 281–293, Springer, New York, 1996. · Zbl 0869.11011
[7] I.Z. Ruzsa, Cardinality questions about sumsets, (to appear). · Zbl 1140.11009
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