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Sum-product formulae

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

This is a survey on sum-product formulae and methods. We state old and new results. Our main objective is to introduce the basic techniques used to bound the size of the product and sumsets of finite subsets of a field.

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Notes

  1. 1.

    A.G.: In my primary school we took N = 12 which was the basic multiple needed for understanding U.K. currency at that time.

  2. 2.

    Erdős claimed that the Supreme Being kept a book of all the best proofs, and only occasionally would allow any mortal to glimpse at “The Book.”

  3. 3.

    A generalized arithmetic progression is the image of a lattice, that is:

    $$\displaystyle{ C:=\{ a_{0} + a_{1}n_{1} + a_{2}n_{2} +\ldots +a_{k}n_{k}:\ 0 \leq n_{j} \leq N_{j} - 1\ \text{for}\ 1 \leq j \leq k\}, }$$

    where \(N_{1},N_{2},\ldots,N_{k}\) are integers ≥ 2. This generalized arithmetic progression is said to have dimension k and volume \(N_{1}N_{2}\ldots N_{k}\); and is proper if its elements are distinct.

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Acknowledgements

Thanks are due to Todd Cochrane, Ernie Croot, Harald Helfgott, Chris Pinner. Andrew Granville is partially supported by an NSERC Discovery Grant, as well as a Canadian Research Chair. Jozsef Solymosi is partially supported by ERC Advanced Research Grant no 267165 (DISCONV), by Hungarian National Research Grant NK 104183, and by an NSERC Discovery Grant.

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Authors and Affiliations

  1. Département de mathématiques et de statistiques, Université de Montréal, CP 6128 Succursale Centre-Ville, Montréal, QC, Canada, H3C 3J7

    Andrew Granville

  2. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T 1Z2

    József Solymosi

Authors
  1. Andrew Granville
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Correspondence to József Solymosi .

Editor information

Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Granville, A., Solymosi, J. (2016). Sum-product formulae. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_18

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