Bounding multiplicative energy by the sumset

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corollary_2_2: Solymosi's sum-product bound: every finite set A of positive real numbers has max(|A+A|, |AA|) >= |A|^(4/3) / (2 ceil(log |A|)^(1/3)), which is the exponent 4/3 up to a logarithmic factor.

lemma_2_3: Solymosi's bound on multiplicative energy by the sumset: every finite set A of positive real numbers has E(A) / ceil(log |A|) <= 4 |A+A|^2, with an asymmetric form for two sets stated in the remarks.

theorem_2_1: Solymosi's main theorem: every finite set A of positive real numbers satisfies |AA| |A+A|^2 >= |A|^4 / (4 ceil(log |A|)), an inequality the paper calls sharp up to the power of the logarithm for A = {1,...,n}.

theorem_3_1: Solymosi's bound for k-fold sumsets of sets with very small product set: for each integer k >= 2 there is delta = delta_k(eps), tending to 0 with eps, such that |AA| <= |A|^(1+eps) implies |kA| >= |A|^(2-1/k-delta).

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József Solymosi, Bounding multiplicative energy by the sumset, Adv. Math. 222 (2009), no. 2, 402--408, doi:10.1016/j.aim.2009.04.006; preprint arXiv:0806.1040. The copy read for this card is arXiv v3 (23 June 2008, 8 pages); labels and pages below are its own, and the journal's page numbers are not mapped.

Solymosi bounds the multiplicative energy $E(A)$ of a finite set of positive reals by the size of its sumset and deduces the inequality $\lvert AA\rvert\,\lvert A+A\rvert^2\ge\lvert A\rvert^4/(4\lceil\log\lvert A\rvert\rceil)$ (Theorem 2.1, p. 2), which the paper calls sharp up to the power of the logarithm for $A=\{1,\ldots,n\}$. Its Corollary 2.2 (p. 2) is the sum-product bound $\max\{\lvert A+A\rvert,\lvert AA\rvert\}\ge\lvert A\rvert^{4/3}/(2\lceil\log\lvert A\rvert\rceil^{1/3})$, improving the earlier exponent $1+3/14$ towards the $2-\varepsilon$ of the Erdős--Szemerédi conjecture (p. 1). The tool is Lemma 2.3 (p. 3), $E(A)/\lceil\log\lvert A\rvert\rceil\le4\lvert A+A\rvert^2$, proved in Section 2.2 (pp. 3--5): $A\times A$ is covered by the $\lvert A/A\rvert$ lines through the origin, a dyadic class of lines, each carrying at least $2^I$ and fewer than $2^{I+1}$ points, carries at least a $1/\lceil\log\lvert A\rvert\rceil$ share of the energy, and the sums of points on consecutive lines of that class are disjoint inside $(A+A)\times(A+A)$. Section 2.3 (p. 5) states an asymmetric form for two sets. Section 3 (pp. 5--7) extends the method to higher dimensions by triangulating the rich lines in $\mathbf{RP}^{k-1}$: Theorem 3.1 (p. 6) shows that $\lvert AA\rvert\le\lvert A\rvert^{1+\varepsilon}$ forces $\lvert kA\rvert\ge\lvert A\rvert^{2-1/k-\delta}$ with $\delta=\delta_k(\varepsilon)\to0$ as $\varepsilon\to0$. The paper does not name the base of its logarithm.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v3; the proofs read for structure only. Nothing here is independently reviewed.

Source: https://arxiv.org/abs/0806.1040. The arXiv record names arXiv's non-exclusive distribution license (arXiv:0806.1040), every other right reserved.

Bears on.

• #818: Theorem 2.1 (p. 2) is proved for finite sets of positive reals; with $\lvert A+A\rvert\le K\lvert A\rvert$ it rearranges to $\lvert AA\rvert\ge\lvert A\rvert^2/(4K^2\lceil\log\lvert A\rvert\rceil)$, the problem's bound with one logarithm. The paper says the theorem shows the product set must be very large when the sumset is small (p. 5); it does not write out the rearrangement, nor the passage to sets of integers that may contain $0$ or negative numbers.
• #52: Corollary 2.2 (p. 2) gives the sum-product exponent $4/3$ up to a logarithmic factor for finite sets of positive reals, which the paper presents as progress on the Erdős--Szemerédi conjecture (p. 1); it does not answer the problem's exponent $2-\epsilon$.

Results.

• Theorem 2.1 (p. 2): for finite $A$ of positive reals, $\lvert AA\rvert\,\lvert A+A\rvert^2\ge\lvert A\rvert^4/(4\lceil\log\lvert A\rvert\rceil)$.
• Corollary 2.2 (p. 2): for finite $A$ of positive reals, $\max\{\lvert A+A\rvert,\lvert AA\rvert\}\ge\lvert A\rvert^{4/3}/(2\lceil\log\lvert A\rvert\rceil^{1/3})$.
• Lemma 2.3 (p. 3): for finite $A$ of positive reals, $E(A)/\lceil\log\lvert A\rvert\rceil\le4\lvert A+A\rvert^2$, with the asymmetric form of Section 2.3 (p. 5).
• Theorem 3.1 (p. 6): for each $k\ge2$, $\lvert AA\rvert\le\lvert A\rvert^{1+\varepsilon}$ implies $\lvert kA\rvert\ge\lvert A\rvert^{2-1/k-\delta}$ with $\delta=\delta_k(\varepsilon)\to0$ as $\varepsilon\to0$.

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