Bounding multiplicative energy by the sumset
type: Item
instance of: scholarly article
title: Bounding multiplicative energy by the sumset
zbMATH Open document ID: 1254.11016
author: József Solymosi
published in: Advances in Mathematics
publication date: 2009-08-21T00:00:00Z
review text: In this short and well-written paper, the author proves some interesting results on the relationship between the sizes of the product set \(AA = \{ ab : a, b \in A \}\) and the sumset \(A+A = \{ a+b : a, b \in A \}\), for a set of positive real numbers \(A\). Specifically, the main result states that \[ |AA||A+A|^2 \geq \frac{ |A|^4}{4 \lceil \log |A| \rceil}, \] which is sharp up to the power of the \(\log\) term, as can be seen by taking \(A = \{1,2,\ldots,n\}\). An immediate consequence is that \[ \max(|AA|,|A+A|) \geq \frac{ |A|^4/3}{2 \lceil \log |A| \rceil^{1/3}}, \] the strongest result known in the direction of the sum-product conjecture of \textit{P. Erdős} and \textit{E. Szemerédi} [Studies in Pure Mathematics, Mem. of P. Turán, 213--218 (1983; Zbl 0526.10011)], which asks, for any \(\varepsilon > 0\), for a bound of the form \[ \max(|AA|,|A+A|) \geq c_\epsilon |A|^{2-\varepsilon} \] with \(c_\varepsilon > 0\). The idea behind this much-studied conjecture is that it should be impossible for a finite set of reals to be multiplicatively and additively structured simultaneously, as the failure of such a bound would suggest. The proof is of a geometric flavour and is entirely elementary. The key step is to establish an upper bound for the so-called multiplicative energy of \(A\), defined as the number of quadruples \((a,b,c,d) \in A^4\) with \(a/b=c/d\), in terms of \(|A+A|\). This is done by producing lots of distinct sums in \((A+A) \times (A+A) = (A \times A) + (A \times A)\), the idea behind which is to consider sums of pairs \((a,b)\), \((c,d)\) with certain carefully chosen ratios \(a/b\), \(c/d\) --- that is, pairs lying on certain lines through the origin. The paper finishes with a lower bound for the size of the iterated sumset \(kA = \{a_1+\cdots+a_k : a_i \in A \}\) when \(AA\) is small: if \(|AA| \leq |A|^{1+\epsilon}\) then \[ |kA| \geq |A|^{2-1/k - \delta}, \] where \(\delta = \delta_k(\epsilon) \to 0\) as \(\epsilon \to 0\). The author notes tha…
reviewed by: Olof Sisask
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 5597906
zbMATH Keywords: Sum-product estimates
Wikidata QID: Q57534457
MaRDI profile type: Publication
full work available at URL: j.aim.2009.04.006
OpenAlex ID: W2029408549
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DOI: 10.1016/J.AIM.2009.04.006
description: scientific article; zbMATH DE number 5597906