Abstract
We investigate numerous cardinality questions concerning sums of finite sets. A typical problem looks like the following: if A has n elements, A + B has cn, what can we deduce about A and B? How can we estimate the cardinalities of other sets like A − B and A + B + A? This is in quest of a generalization of Freiman’s famous theorem that describes the structure of those sets A for which A + A is small, to the case of different summands.
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References
Brailovsky, L.V., Freiman, G. A., ‘On a product of finite subsets in a torsion-free group’, J. Algebra130(1990), 462–476.
Freiman, G. A ., Foundations of a structural theory of set addition (in Russian), Kazan Gos. Ped. Inst., Kazan 1966.
Freiman, G. A ., Foundations of a structural theory of set addition, Translation of Mathematical Monographs vol. 37, Amer. Math. Soc., Providence, E. I., USA 1973.
Freiman, G. A, ‘What is the structure of K if K + K is small?’, in: Lecture Notes in Mathematics 1240, Springer-Verlag, New York-Berlin (1987), 109–134.
Freiman, G. A., Pigaev,’V. P., ‘The relation between the invariants R and T (Russian)’, Kalinin. Gos. Univ. Moscow (1973), 172–174.
Halberstam, H.; Roth, K. F, Sequences, Clarendon, London; (2nd ed. Springer-Verlag, New York-Berlin, 1983) 1966.
Kemperman, J. H. B ., ‘On complexes in a semigroup’, Indag. Math. 18(1956), 247–254.
Pliinnecke, EL, ‘Eine zahlentheoretische Anwendung der Graphtheorie’, J. Reine Angew. Math. 243(1970)
Ruzsa, I. Z., ‘On the cardinality of A + A and A - A’ in: Coll. Math. Soc. Bolyai 18, Combinatorics (Keszthely 1976), Akadémiai Kiadô, Budapest (1979), 933-938.
Ruzsa, I. Z., ‘Sets of sums and differences’, in: Séminaire de Théorie des Nombres, Paris 1982/83, Birkhâuser(1984), 267–273.
Ruzsa, I. Z., ‘An application of graph theory to additive number theory’, Scientia, Ser. A 3(1989),97–109.
Ruzsa, I. Z., ‘On the number of sums and differences’, Acta Math. Sci. Hungar., to appear.
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© 1996 Springer-Verlag New York, Inc.
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Ruzsa, I.Z. (1996). Sums of Finite Sets. In: Chudnovsky, D.V., Chudnovsky, G.V., Nathanson, M.B. (eds) Number Theory: New York Seminar 1991–1995. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2418-1_21
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DOI: https://doi.org/10.1007/978-1-4612-2418-1_21
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