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Around the Sum-product Phenomenon

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Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

Abstract

The purpose of this exposé is to give a sample of P. Erdős many contributions to combinatorics that turned out to be unexpectedly seminal. He was indeed a master in recognizing seemingly elementary questions which require new insights, often with far reaching consequences. The results discussed below originate from his papers “Problems and results in combinatorial number theory, III” ([32]), “On sums and products of integers”, ([33]), jointly with Szemerédi, and “Additive Gruppen mit vorgegebener Hausdorffscher Dimension” ([34]), jointly with Volkmann. These papers led to numerous developments over the past decade and influenced other parts of mathematics, including number theory, theoretical computer science, ergodic theory and group theory. Giving a fair account of them would be a considerable task and we limit ourselves to citing just a few. The choice only reflects the author’s interests and research; many related contributions and contributors will not be cited and the bibliography strictly serves this presentation.

The research was partially supported by NSF grants DMS-0808042 and DMS-0835373.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Jean Bourgain

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  1. Jean Bourgain
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Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

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(Dedicated to P. Erdős)

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© 2013 János Bolyai Mathematical Society and Springer-Verlag

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Bourgain, J. (2013). Around the Sum-product Phenomenon. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_4

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