2003/08/01 by György Elekes Gy., Imre Z. Ruzsa · 5 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Computational Geometry and Mesh Generation #Limits and Structures in Graph Theory
paper · doi:10.1556/sscmath.40.2003.3.4
openalex publication_date 2003/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
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 n 2 / (a 4 log n ), and the number of quotients is = c n 2 / min (a 6 , a 4 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 n 2 / (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.