A low-energy decomposition theorem

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theorem_1_1: Balog and Wooley's low-energy decomposition: every finite set A of reals is the disjoint union of B and C with E_+(B) and E_x(C) both at most a constant times |A|^(3-2/33)(log |A|)^(31/33), and with the additive and multiplicative energies between B and C at most a constant times |A|^(3-1/33)(log |A|)^(31/66).

theorem_1_2: Balog and Wooley's bounds 1/3 <= kappa <= 31/33 for the infimum kappa of the permissible low-energy decomposition exponents, with the lower bound from an integer set of the form (2m-1)2^n in which every subset of at least half the set has additive and multiplicative energy of order N^(7/3).

theorem_1_3: Balog and Wooley's decomposition over F_p: for a large prime p and A in F_p with |A| at most p^(101/161)(log p)^(71/161), A splits into B and C with E_+(B) and E_x(C) at most a constant times |A|^(3-4/101)(log |A|)^(1-2/101), with a weaker bound involving (|A|/p)^(1/15) for larger A.

theorem_1_4: Balog and Wooley's k-fold form of the low-energy decomposition: for integers m, n >= 2 every finite real set A splits into B and C with the m-fold additive energy of B at most a constant times |A|^(2m-1-2/33)(log |A|)^(31/33) and the n-fold multiplicative energy of C at most a constant times |A|^(2n-1-2/33)(log |A|)^(31/33).

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Antal Balog, Trevor D. Wooley, "A low-energy decomposition theorem," Quart. J. Math. 68 (2017), no. 1, 207--226, DOI 10.1093/qmath/haw023. The copy read for this card is the arXiv preprint, arXiv:1510.03309v1 (12 October 2015), which prints no notice; the arXiv abstract page names arXiv's non-exclusive distribution license (https://arxiv.org/abs/1510.03309v1, read 2026-10-02), every other right reserved.

For finite sets of reals, write $E_+(A)$ and $E_\times(A)$ for the numbers of additive and multiplicative quadruples in $A$. Theorem 1.1 (p. 2; proof in Section 3, pp. 6--10) proves that every finite $A\subset \mathbb R$ has a disjoint decomposition $A=B\cup C$ such that, with $\delta=2/33$,

$$\max\{E_+(B),E_\times(C)\} \ll |A|^{3-\delta}(\log |A|)^{1-\delta},$$

and also

$$\max\{E_+(B,C),E_\times(B,C)\} \ll |A|^{3-\delta/2}(\log |A|)^{(1-\delta)/2}.$$

Thus one part is quantitatively non-additive, the other quantitatively non-multiplicative, and the two parts have small mutual energy in both operations. The introductory example (1.2), on p. 2, first explains why no assertion about the smaller of $E_+(A)$ and $E_\times(A)$ can hold without a decomposition: an arithmetic progression joined to a geometric progression has both energies of order $|A|^3$.

Decomposition mechanism

Starting with the residual set $B_k$, the proof stops once $E_+(B_k)\le N^{3-\delta}(\log N)^\theta$. Otherwise the quantitative Balog--Szemeredi--Gowers consequence in Lemma 3.3 extracts a large subset $A_{k+1}\subset B_k$ with small additive doubling, namely (3.3)--(3.4):

$$|A_{k+1}|\gg N^{1-3\delta/4}(\log N)^{(3\theta-5)/4}, \qquad |A_{k+1}+A_{k+1}| \ll N^{7\delta}(\log N)^{5-7\theta}|A_{k+1}|.$$

The extracted pieces are removed from the residual set and accumulated in $C$. Their minimum size forces termination after at most $K_0$ steps. The proof then groups them by dyadic cardinality, applies the union-energy bound in Lemma 3.4 and Solymosi's mixed-energy estimate in Lemma 3.5, and obtains

$$E_\times(C)\ll N^{2+31\delta/2}(\log N)^{31(1-\theta)/2}.$$

Balancing this with the stopping threshold gives $\delta=2/33$ and $\theta=31/33$. Cauchy--Schwarz applied to difference and ratio representation functions then supplies the two cross-energy bounds at the end of Section 3.

The one-dimensional prototype and its limit

Theorem 1.2 (p. 3; construction and proof in Section 2, pp. 5--6) defines a permissible decomposition exponent $\beta$ by asking for $\max\{E_+(B),E_\times(C)\}\le |A|^{2+\beta+\varepsilon}$, and proves that its infimum $\kappa$ satisfies

$$\frac13\le \kappa\le\frac{31}{33}.$$

The lower bound comes from the explicit integer set at the opening of Section 2,

$$A_N=\{(2m-1)2^n:m\le N^{2/3},\ n\le N^{1/3}\}, \qquad |A_N|=N+O(N^{2/3}).$$

This is $A_N=GP$, with $G$ a geometric progression of powers of $2$ and $P$ a progression of odd integers. Every subset containing at least half of $A_N$ has both energies $\gg N^{7/3}$, as recorded in (2.1). For multiplication this follows from $|B\cdot B|\le 4N^{5/3}$ and Cauchy--Schwarz; for addition, the proof finds $\gg N^{1/3}$ dense fixed-$n$ layers, each contributing $\gg N^2$ additive quadruples. Hence every partition has either $E_+(B)\gg N^{7/3}$ or $E_\times(C)\gg N^{7/3}$, ruling out every $\beta<1/3$.

This energy obstruction is not a sum-product counterexample; the paper does not discuss $A_N+A_N$, and the argument below is this card's own. Put $M=\lfloor N^{2/3}\rfloor$, $L=\lfloor N^{1/3}\rfloor$, and $D=\lceil\log_2(2M)\rceil$. Among sums

$$(2m_1-1)2^{n_1}+(2m_2-1)2^{n_2} \quad(n_2-n_1\ge D),$$

the $2$-adic valuation recovers $n_1$. If two such sums are equal, divide by $2^{n_1}$ and reduce modulo $2^D$: the two lower odd coefficients are congruent modulo $2^D$, and their difference has absolute value below $2M\le2^D$, so they are equal. The remaining terms are then equal, and unique factorization gives equal exponent gaps and equal upper odd coefficients. These $M^2(L-D)(L-D+1)/2=\Omega(N^2)$ sums are distinct. Thus $|A_N+A_N|\gg |A_N|^2$: the same concentration that makes the layerwise additive energy large does not make the whole sumset small.

Section 2 of Bloom--Sawin--Schildkraut--Zhelezov explicitly calls its construction a high-dimensional version of this standard Balog--Wooley example. It says that the simplest one-dimensional $GP$ makes both $|A+A|$ and $|AA|$ smaller than $|A|^2$ only by a logarithmic factor, and that $|A+A|\ge |A|^{2-o(1)}$ still holds because the geometric progression is exponentially sparse. BSSZ replace $G$ by a dense box in the unit lattice of a high-degree totally real number field and $P$ by a high-dimensional additive lattice box. That change produces a power saving in both the real sumset and product set. Its sets are real algebraic integers, not rational integers, and the integer statement of Problem 52 is a separate question.

Finite fields and k-fold energies

Theorem 1.3 (p. 3; proof in Section 4, pp. 10--14) gives a decomposition of $A\subseteq\mathbb F_p$ with $\delta=4/101$ when $|A|\le p^{101/161}(\log p)^{71/161}$ and a bound with the factor $(|A|/p)^{1/15}$ for larger $A$; (1.3) shows that no bound uniform in $p$ is possible. Theorem 1.4 (p. 4; proof in Section 5, pp. 14--15) transfers Theorem 1.1 to $m$-fold additive and $n$-fold multiplicative energies with the same saving $|A|^{2/33}$.

Read status. Claims checked: Theorems 1.1--1.4, the definition of a permissible exponent, the construction of Section 2 with (2.1), and (1.3) were read clause by clause on the preprint's pages. The proofs were read but not checked step by step.

Bears on. #52: the paper notes (pp. 2--3) that decompositions with $\max\{E_+(B),E_\times(C)\}\ll|A|^{2+\varepsilon}$ would imply the Erdős--Szemerédi bound through (1.1); Theorem 1.2 shows they do not exist in general, by an example made of integers, so that route to the problem is closed. Theorem 1.1 itself yields only $\max\{|A+A|,|A\cdot A|\}\gg|A|^{1+2/33}(\log|A|)^{-31/33}$ (an observation of the result page), weaker than Solymosi's bound quoted on p. 1. Nothing here settles the problem.

Results. Theorem 1.1 (p. 2); Theorem 1.2 (p. 3, with the construction and (2.1) of Section 2, p. 5); Theorem 1.3 (p. 3, with Theorem 4.2, p. 10); Theorem 1.4 (p. 4). Lemmas 3.1--3.5 (pp. 6--8) and Lemmas 4.3--4.6 (p. 11) are proof steps, summarized on the pages of the theorems they serve; Lemma 4.1 (p. 10) serves only the earlier Theorem 4.2.

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