Abstract
The aim of this paper is to prove a general version of Plünnecke’s inequality. Namely, assume that for finite sets A, B 1 ,…,B k we have information on the size of the sumsets A+Bi 1+…+Bi l for all choices of indices i 1,…,i l . Then we prove the existence of a non-empty subset X of A such that we have good control’ over the size of the sumset X+B 1+…+B k . As an application of this result we generalize an inequality of [1] concerning the submultiplicativity of cardinalities of sumsets.
Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 43631, T 43623, T 49693.
Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. PF-64061, T-049301, T-047276
Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 43623, T 42750, K 61908.
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References
K. Gyarmati, M. Matolcsi and I. Z. Ruzsa, A superadditivity and submultiplicativity property for cardinalities of sumsets, Combinatorica, to appear (also arXiv:0707.2707v1).
J. L. Malouf, On a theorem of Plünnecke concerning the sum of a basis and a set of positive density, J. Number Theory, 54.
M. B. Nathanson, Additive number theory: Inverse problems and the geometry of sumsets, Springer, 1996.
H. Plünnecke, Eine zahlentheoretische Anwendung der Graphtheorie, J. Reine Angew. Math., 243 (1970), 171–183.
I. Z. Ruzsa, An application of graph theory to additive number theory, Scientia, Ser. A, 3 (1989), 97–109.
I. Z. Ruzsa, Addendum to: An application of graph theory to additive number theory, Scientia, Ser. A, 4 (1990/91), 93–94.
T. Tao and V. H. Vu, Additive combinatorics, Cambridge University Press, Cambridge, 2006.
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© 2008 János Bolyai Mathematical Society and Springer-Verlag
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Gyarmati, K., Matolcsi, M., Ruzsa, I.Z. (2008). Plünnecke’s Inequality for Different Summands. In: Grötschel, M., Katona, G.O.H., Sági, G. (eds) Building Bridges. Bolyai Society Mathematical Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85221-6_10
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DOI: https://doi.org/10.1007/978-3-540-85221-6_10
Publisher Name: Springer, Berlin, Heidelberg
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