on the sum-product conjecture's falsity

Overview

Bloom, Sawin, Schildkraut, and Zhelezov disprove the real sum-product conjecture in its full-strength form (Bloom et al., 2026). They construct arbitrarily large finite sets $A\subset \mathbb{R}$ for which both the sum set and product set are power-saved below the trivial quadratic size:

$$\max (|A+A|,|AA|)\le |A{|}^{2-c}$$

for an absolute constant $c>0$. The construction is not a counterexample in $\mathbb{Z}$: the elements live in totally real number fields whose degree grows like $\asymp \log |A|$.

🏷️ The Conjecture and the Break

For a finite set $A$ in a ring, write

$$A+A=\{a+b:a,b\in A\},AA=\{ab:a,b\in A\}.$$

The sum-product conjecture predicts that one of these sets must be almost quadratically large:

$$\max (|A+A|,|AA|)\ge |A{|}^{2-o(1)}.$$

The paper proves this is false over $\mathbb{R}$. More precisely, there is an absolute $c>0$ and arbitrarily large $A\subset \mathbb{R}$ with

$$\max (|A+A|,|AA|)\le |A{|}^{2-c}.$$

The important caveat is algebraic: the construction embeds the ring of integers ${\mathcal{O}}_K$ of a totally real number field $K$ into $\mathbb{R}$. The degree $d=[K:\mathbb{Q}]$ tends to infinity with $|A|$. Thus the result leaves open the classical integer version, and also leaves room for a bounded-degree number-field version.

Quantitative form

There is an absolute constant $C>0$ such that for infinitely many degrees $d$ there are totally real number fields $K/\mathbb{Q}$ of degree $d$ with the following property. For every $X\ge 1$ there is $A\subset {\mathcal{O}}_K$ such that

$$X^d\le |A|\le (CX{)}^d,$$

and

$$|A+A|\le C^d|A|,|AA|\le 2^{-d}|A{|}^2.$$

Choosing $X$ large relative to $C$ turns the factor $C^d$ in the sum-set into $|A{|}^{\epsilon }$, while the factor $2^{-d}$ becomes a fixed power saving in the product set.

🏷️ Preliminaries: Small Growth

If $|A|=n$, then the largest possible sizes of $A+A$ and $AA$ are both on the order of $n^2$. This is what one expects when almost all pairs give different sums and products. The sum-product problem asks whether a finite real set must expand almost this much in at least one of the two operations.

Small sum-set and small product-set mean different kinds of structure.

Additive structure

If

$$P=\{1,2,\dots ,N\},$$

then

$$P+P=\{2,3,\dots ,2N\},$$

so $|P+P|=2N-1$, which is linear in $|P|$ rather than quadratic. Arithmetic progressions are the basic model for additive structure.

Multiplicative structure

If

$$G=\{1,r,r^2,\dots ,r^{M-1}\},$$

then

$$GG=\{1,r,r^2,\dots ,r^{2M-2}\},$$

so $|GG|=2M-1$. Geometric progressions are the basic model for multiplicative structure.

The tension is that an arithmetic progression usually has a large product set, while a geometric progression usually has a large sum-set. The conjectural principle was: a real set cannot be strongly arithmetic and strongly geometric at the same time. The new paper shows that this principle fails when one is allowed to use algebraic numbers of unbounded degree.

🏷️ The Balog-Wooley Toy Model

The construction should be read as a high-dimensional upgrade of a Balog-Wooley type example (Balog & Wooley, 2017). Start with

$$A=GP,$$

where $G$ is multiplicatively structured and $P$ is additively structured. One should think of $P$ as an interval and $G$ as a short geometric progression.

Why does this help? Multiplication by elements of $G$ creates several scaled copies of $P$. The set $A$ may have size about $|G||P|$, but its product set satisfies

$$AA\subseteq GGPP.$$

If $GG$ is small, then the product set loses roughly one factor of $|G|$ compared with the trivial $|A{|}^2$ bound.

The sum-set is controlled in a different way. If all elements of $GP$ lie in a not-too-long interval, then $GP+GP$ also lies in a not-too-long interval. This is the one-dimensional ancestor of the later lattice-box estimate.

Why the toy model does not prove the theorem

In $\mathbb{Z}$, the multiplicative set $G$ is typically a geometric progression. Such a progression has good product structure, but it is sparse: its largest element grows exponentially in its length. That exponential spread makes the interval containing $GP+GP$ too long. The product set gains a factor from $G$, but the sum-set pays for the height of $G$.

The breakthrough is to keep the multiplicative structure of $G$ without paying the one-dimensional sparsity cost. Number fields supply a replacement: the unit group of a degree-$d$ field has rank $d-1$, so it contains many multiplicatively structured elements inside a bounded logarithmic box.

🏷️ Number-Field Dictionary

A number field $K$ is a finite extension of $\mathbb{Q}$. It is totally real if every embedding of $K$ into $\mathbb{C}$ lands in $\mathbb{R}$. If $[K:\mathbb{Q}]=d$, then there are $d$ real embeddings

$${\sigma }_1,\dots ,{\sigma }_d:K\hookrightarrow \mathbb{R}.$$

The ring of integers ${\mathcal{O}}_K$ is the analogue of $\mathbb{Z}$ inside $K$. The construction uses two structures on the same set of algebraic integers.

Additive lattice: the replacement for an interval

The Minkowski embedding sends

$$\alpha \mapsto ({\sigma }_1(\alpha ),\dots ,{\sigma }_d(\alpha )).$$

This identifies ${\mathcal{O}}_K$ with a full-rank lattice in ${\mathbb{R}}^d$. Counting algebraic integers with bounded embeddings is therefore a geometry-of-numbers problem. The box

$$B^{+}(X)=\{\alpha \in {\mathcal{O}}_K:|{\sigma }_i(\alpha )|\le X\text{for all }i\}$$

plays the role of a $d$-dimensional interval.

Units: the replacement for a geometric progression

A unit is an algebraic integer $u\in {\mathcal{O}}_K$ whose inverse is also in ${\mathcal{O}}_K$. Dirichlet’s unit theorem says that, after ignoring signs, the units form a rank $d-1$ lattice under the logarithmic embedding

$$u\mapsto (\log |{\sigma }_1(u)|,\dots ,\log |{\sigma }_d(u)|).$$

The image lies in the hyperplane

$$H=\{x_1+\cdots +x_d=0\},$$

because a unit has norm $\pm 1$, so the product of its embeddings has absolute value $1$.

The multiplicative box is

$$B^{\times }(Y)=\{u\in {\mathcal{O}}_K^{\times }:|\log |{\sigma }_i(u)||\le Y\text{for all }i\}.$$

This is the high-rank analogue of a short geometric progression. The key difference is density: for fixed $Y$, this box can contain exponentially many units as $d$ grows, because the unit lattice has rank $d-1$.

Discriminant and regulator

The discriminant ${\Delta }_K$ controls the covolume of the additive lattice ${\mathcal{O}}_K\subset {\mathbb{R}}^d$. The regulator $R_K$ controls the covolume of the logarithmic unit lattice. Small covolume means many lattice points in boxes. The paper needs fields with

$${\Delta }_K\le C^d$$

for infinitely many $d$, and also needs $R_K$ bounded exponentially in $d$. Martinet’s class-field-tower construction supplies the discriminant bound, and the paper uses the elementary estimate

$$R_K\le {\Delta }_K$$

for totally real $K$ of degree at least $2$.

🏷️ The Set $A=GP$

Now the Balog-Wooley template is implemented inside ${\mathcal{O}}_K$.

The additive component is a translated additive box

$$P=X+B^{+}(\epsilon X),$$

so every embedding of every $p\in P$ lies close to $X$. This is the number-field version of taking an interval.

The multiplicative component is a unit box

$$G=B^{\times }(Y),$$

so every $u\in G$ changes each embedding by at most a factor $e^Y$. This is the number-field version of taking a short geometric progression, but with rank $d-1$ rather than rank $1$.

The final set is

$$A=GP=\{up:u\in G,p\in P\}.$$

The product map $G\times P\to A$ is injective once $\epsilon$ is small. Indeed, if

$$u_1{p}_1=u_2{p}_2,$$

then

$$u_1/u_2=p_1/p_2$$

is a unit whose every embedding is very close to $1$. A separation lemma for units says that if a unit has all embeddings inside $({\varphi }^{-1},\varphi )$ in absolute value, where $\varphi =(1+\sqrt{5})/2$, then the unit is $\pm 1$. The sign is forced to be $+1$ by the positivity of the embeddings of $p_1/p_2$, so $u_1=u_2$ and then $p_1=p_2$.

Hence

$$|A|=|G||P|.$$

🏷️ Why the Sum Set Is Small

This is the additive half of the Balog-Wooley idea. Instead of saying that $A+A$ lies in a short interval, we say that all its embeddings lie in a controlled $d$-dimensional box.

Every $a=up\in A$ has all real embeddings bounded by about $X{e}^Y$. Therefore

$$A+A\subseteq B^{+}(CX{e}^Y)$$

for an absolute constant $C$. Counting lattice points in this additive box gives

$$|A+A|\le (C{e}^Y{)}^d{\Delta }_K^{1/2}|A|.$$

Using ${\Delta }_K\le C_0^d$, this becomes

$$|A+A|\le C_1^d{e}^{Yd}|A|.$$

Once $Y$ is fixed and $X$ is chosen large enough, $|A|\asymp X^d$ is large enough to absorb this exponential-in-$d$ loss:

$$|A+A|\le |A{|}^{1+\epsilon }.$$

🏷️ Why the Product Set Is Small

This is the multiplicative half. The set $P$ is not supposed to have a small product set; it is the unit box $G$ that supplies the saving.

The product set satisfies

$$AA\subseteq GGPP.$$

The additive box is estimated trivially:

$$|PP|\le |P{|}^2.$$

The unit box has genuine multiplicative structure:

$$GG\subseteq B^{\times }(2Y),$$

so the number of possible unit products is still only exponential in $d$ times $|G|$:

$$|GG|\le C^d|G|.$$

Combining these estimates and using $|A|=|G||P|$ gives the schematic inequality

$$|AA|\le \frac{C^d}{|G|}|A{|}^2.$$

Since $|G|$ is on the order of $Y^{d-1}$ up to exponential-in-$d$ constants, choosing $Y$ as a sufficiently large absolute constant forces

$$|AA|\le 2^{-d}|A{|}^2.$$

Because $|A|\asymp X^d$, the factor $2^{-d}$ is a fixed power saving in $|A|$ after $X$ is fixed large enough.

Proof spine

The construction balances three scales. The degree $d$ is the source of the eventual power saving. The unit-box width $Y$ is fixed large enough so that $|G|$ beats the exponential constants in the product-set estimate. The additive-box scale $X$ is then chosen large enough so that the exponential constants in the sum-set estimate are negligible compared with $|A{|}^{\epsilon }$. Martinet’s tower supplies arbitrarily large $d$ while keeping ${\Delta }_K$ and $R_K$ under exponential control.

🏷️ Many Sums and Products

The same mechanism also defeats the many sums and products conjecture over $\mathbb{R}$. For fixed $k\ge 3$, the paper constructs arbitrarily large $A\subset \mathbb{R}$ with

$$\max (|kA|,|A^{(k)}|)\le |A{|}^{C\frac{\log k}{\log \log k}}.$$

Here $kA$ is the $k$-fold sum-set and $A^{(k)}$ is the $k$-fold product set. The construction is even simpler: take

$$A=B^{\times }(Y)$$

itself. Then

$$A^{(k)}\subseteq B^{\times }(kY),$$

and

$$kA\subseteq B^{+}(k{e}^Y).$$

The same lattice counts control both sides.

🏷️ Linear Equations and Other Fields

The construction has consequences beyond the real sum-product problem.

Linear equations in multiplicative groups

For sufficiently large $k$, the authors construct rank-$d$ multiplicative groups $\Gamma \le {\mathbb{R}}^{\times }$ with at least

$$(Ck{)}^{kd}$$

positive solutions to

$$x_1+\cdots +x_k=1,x_i\in \Gamma .$$

This shows that the linear dependence on $d$ in the exponent of the Evertse-Schlickewei-Schmidt upper bound is best possible for subgroups of ${\mathbb{R}}^{\times }$.

Variants

The paper also gives analogues in ${\mathbb{Q}}_p$, in finite fields ${\mathbb{F}}_p$, and in Laurent series fields ${\mathbb{F}}_q((t))$. The finite-field result produces reasonably large sets $A\subset {\mathbb{F}}_p$ with $p^c<|A|<p^{1/2}$ and

$$\max (|A+A|,|AA|)\le |A{|}^{2-c}.$$

In positive characteristic local fields, the exponent saving has the form $c/\log p$, and in small characteristic the paper gives explicit exponents near $1.9$.

🏷️ What the Counterexample Means

This result does not say that sum-product expansion is weak for typical real sets. It says that the real field contains increasingly high-degree algebraic shadows of structured number fields. Inside those shadows, the additive geometry of ${\mathcal{O}}_K$ and the multiplicative geometry of ${\mathcal{O}}_K^{\times }$ can be made simultaneously visible.

The old heuristic that a set cannot be additively and multiplicatively structured at the same time remains a good guide in fixed algebraic complexity. The construction escapes by increasing the ambient algebraic degree with the size of the set. The additive structure lives in the Minkowski lattice of algebraic integers, while the multiplicative structure lives in the logarithmic lattice of units; unbounded degree lets both lattices have enough room at once.

Remaining boundary

The integer case $A\subset \mathbb{Z}$ is not touched. Nor does the construction rule out a version of the conjecture for subsets of a fixed number field. The failure occurs in $\mathbb{R}$ because $\mathbb{R}$ contains totally real fields of unbounded degree.

🔗 See Also

• on sum-product in finite fields via entropy --- This note studies sum-product growth in finite fields from the entropy side; the present construction instead uses algebraic number fields and unit lattices to force simultaneous small additive and multiplicative growth.
• on new lower bounds of R(3, k) --- Both papers are examples of recent counter-conjectural constructions where a structured high-dimensional object beats a long-standing random-process or expansion heuristic.
• on the lower bound of Ramsey number --- The comparison is methodological: both settings turn an extremal lower-bound problem into the design of a set or graph whose apparent randomness is constrained by hidden algebraic structure.

📚 References

🐻  Balog, A. & Wooley, T.D. 2017. A low-energy decomposition theorem. The Quarterly Journal of Mathematics 68(1), 207–226.

🐻  Bloom, T.F., Sawin, W., Schildkraut, C. & Zhelezov, D. 2026. The sum-product conjecture is false for real numbers.