Bounding multiplicative energy by the sumset
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 that for sets of integers a much stronger result due to \textit{M.-C. Chang} [Ann. Math. (2) 157, No. 3, 939--957 (2003; Zbl 1055.11017)] is known.
scientific article; zbMATH DE number 5597906