Few sums, many products
Few sums, many products
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金额少，产品多
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发表时间：
2003
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通讯作者：
I. Ruzsa
中科院分区：
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作者：
György Elekes Gy.;I. Ruzsa
设A是n个实数的集合，使得不同的双重和的个数为A n。我们证明了双重乘积的个数为= c n2/ (a4 log n)，商的个数为= c n2/ min (a6, a4 log n)，带有某个绝对常数c。对于有界的A，这给出了商的正确数量级。根据Pomerance和Sarkozy的建议，我们认为和的正确顺序是n2/ (log n)a，其中a<1，可能是2 log 2 -1。我们还给出了由不同集合构成的和、积和商的更一般的不等式。证明使用几何工具，主要是szemeredii - trotter不等式。
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 Sarkozy suggests. We also give more general inequalities for sums, products and quotients formed with different sets. The proofs use geometric tools, mainly the Szemeredi-Trotter inequality.