Let A be a set of n real numbers such that the number of distinct twofold sums is αn. We show that the number of twofold products is ≧ cn2/(α4 log n), and the number of quotients is ≧ cn2/min (α6, α 4 log n) with some absolute constant c. For bounded α 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.
- Product set
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