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  • 1 Eötvös Loránd Tudományegyetem, Természettudományi Kar, Számítógéptudományi Tanszék Pázmány Péter Sétány 1/C, H-1117 Budapest, Hungary
  • 2 Magyar Tudományos Akadémia, Rényi Alfréd Matematikai Kutatóintézet Postafiók 127, H-1364 Budapest, Hungary
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Let A be a set of n real numbers such that the number of distinct twofold sums is a n. We show that the number of twofold products is = c n2/ (a4 log n), and the number of quotients is = c n2/ min (a6, a4 log n) with some absolute constant c. For bounded a this gives the correct order of magnitude for the quotients. For sums we think that the correct order is n2/ (log n)a with some a<1, perhaps with 2 log 2 -1, as a result of Pomerance and Sárközy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemerédi-Trotter inequality.

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