# The Erdős–Szeméredi problem on sum set and product set | Annals of Mathematics

Abstract The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erdős-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A+A small and then deriving that the product set AA is large (using Freiman’s structure theorem) (cf. [N-T], [Na3]). We follow the reverse route and prove that if |AA|c′|A|2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plünnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in k. If g(k)≡min{|A[1]|+|A{1}|} over all sets A⊂ℤ of cardinality |A|=k and where A[1] (respectively, A{1}) refers to the simple sum (resp., product) of elements of A. (See (0.6), (0.7).) It was conjectured in [E-S] that g(k) grows faster than any power of k for k→∞. We will prove here that lng(k)∼(lnk)2lnlnk (see Theorem 2) which is the main result of this paper.

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### 数学年刊（Annals of Mathematics）

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