🔢 mathematics

Both $AA$ and $(A+1)(A+1)$ can be small

This paper demonstrates the existence of arbitrarily large finite sets of real numbers where both the product set $AA$ and the shifted product set $(A+1)(A+1)$ are simultaneously significantly smaller than the trivial quadratic bound, thereby disproving a strong form of the sum-product conjecture over $\mathbb{R}$ .

Original authors: Oliver Roche-Newton, Audie Warren

Published 2026-06-24

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Oliver Roche-Newton, Audie Warren

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant box of numbers. In the world of mathematics, there's a famous rule of thumb called the "Sum-Product Conjecture." It basically says: You can't have it both ways.

If you take a group of numbers and mix them together by adding them (like making a sum), you usually get a huge, messy pile of new numbers. If you mix them by multiplying them, you also get a huge, messy pile. The conjecture claimed that you could never find a special group of numbers where both the addition-mixing and the multiplication-mixing stayed small and tidy at the same time. It was believed that if you kept the multiplication small, the addition would explode, and vice versa.

This paper, written by Oliver Roche-Newton and Audie Warren, is like a magician pulling a rabbit out of a hat to say: "Actually, you can have it both ways."

Here is the simple breakdown of how they did it:

The Setup: A Special Mathematical Playground

To prove their point, the authors didn't just pick random numbers like 1, 2, or 3. They built a very specific, high-tech "playground" using a concept from advanced math called a Number Field.

Think of this Number Field as a special universe with its own rules. Inside this universe, they created two distinct types of "clumps" of numbers:

The "Additive" Clump: A group of numbers that, when you add them together, stay very close to each other (like a tight-knit family).
The "Multiplicative" Clump: A group of numbers that, when you multiply them, also stay very close to each other (like a set of gears that fit perfectly together).

The Trick: The "Shadow" and the "Mirror"

The authors combined these two clumps to create their final set, let's call it Set A.

The Multiplication Test: When they multiplied every number in Set A by every other number in Set A, the result was surprisingly small. It was like taking a small handful of marbles and rolling them together, but they only landed in a tiny, specific corner of the floor.
The Addition Test (The Twist): Here is the tricky part. They didn't just add the numbers; they added 1 to every number first (shifting the whole group), and then multiplied them.
- Imagine you have a line of people. You ask them to multiply their names (weird, right?). Then, you ask them to all take one step forward, and then multiply their new positions.
- Usually, taking that step forward (the "shift") ruins the pattern. It scatters the results.
- But in this paper's construction, the shift didn't break the pattern. The authors showed that even after shifting the numbers by 1, the resulting multiplication still stayed small and tidy.

The "Many Translates" Bonus

The paper goes a step further. They showed that this trick doesn't just work for shifting by 1. You can shift the numbers by any specific set of integers (like 1, 5, or 100), and as long as you build your "playground" big enough, the multiplication will still stay small.

The Big Picture

Before this paper, mathematicians believed that if you had a set of numbers where multiplication was "small," adding a shift (like +1) would force the multiplication to become "huge."

This paper proves that belief wrong. They constructed a set where:

The numbers multiply to a small result.
The numbers shifted by 1 also multiply to a small result.

The Analogy:
Imagine a dance floor.

The Old Belief: If you tell everyone to dance in a tight circle (multiplication), and then you tell them to all take one step to the right (the shift), the circle will break, and they will scatter all over the room.
This Paper's Discovery: They found a way to choreograph the dance so that even after everyone takes that step to the right, they still end up in a tight circle.

Conclusion

The authors have shown that there is a "loophole" in the old rules. They constructed a specific, finite set of real numbers where the maximum size of the product set and the shifted product set is significantly smaller than anyone thought possible.

Important Note: The authors used a generative AI to help brainstorm the ideas for this note, but they explicitly state that every word written and every mathematical step verified was done by humans. They are not claiming this works for medical treatments or real-world engineering yet; it is a pure mathematical discovery about how numbers can behave in unexpected ways.

Technical Summary: "Both AA and (A + 1)(A + 1) Can Be Small"

Problem Statement
The paper addresses a central question in additive combinatorics regarding the interplay between additive and multiplicative structures within subsets of the real numbers ( $\mathbb{R}$ ). A fundamental principle of sum-product theory posits that additive shifts disrupt multiplicative structure. Specifically, it is generally expected that for a finite set $A \subset \mathbb{R}$ , at least one of the product set $AA$ or the shifted product set $(A+1)(A+1)$ must be large.

The best known lower bound for this maximum, established by Stevens and Warren, states that $\max\{|AA|, |(A+1)(A+1)|\} \geq |A|^{1 + \frac{11}{38} - o(1)}$ . A folklore conjecture suggested that the exponent in this inequality could be taken arbitrarily close to 2, implying that it is impossible for both sets to be simultaneously small (i.e., of size $O(|A|^{2-\epsilon})$ ). However, recent work by Bloom, Sawin, Schildkraut, and Zhelezov (BSZ) refuted the standard sum-product conjecture over $\mathbb{R}$ , suggesting that the exponent 2 might not be the correct barrier. This paper investigates whether the specific conjecture regarding the simultaneous smallness of $AA$ and $(A+1)(A+1)$ holds true.

Methodology
The authors adapt the construction used by BSZ to disprove the general sum-product conjecture. The methodology relies on number-theoretic objects within a totally real number field $K$ of degree $d$ over $\mathbb{Q}$ . The construction proceeds as follows:

Lattice Construction: Utilizing a theorem by Martinet, the authors select a totally real number field $K$ with a discriminant $\Delta_K$ bounded by $C^d$ . They consider the ring of algebraic integers $\mathcal{O}_K$ and its group of units $\mathcal{O}_K^*$ .
Geometric Embeddings:
- Additive Structure: The Minkowski embedding $\phi: \mathcal{O}_K \to \mathbb{R}^d$ maps algebraic integers to a lattice $\Lambda$ . The authors define a set $B^+(R)$ of algebraic integers whose embeddings lie within a box of side length $2R$ . The sumset of this set is small because it remains contained within a slightly larger box.
- Multiplicative Structure: The logarithmic embedding $\psi: \mathcal{O}_K^* \to \mathbb{R}^d$ maps units to a lattice of rank $d-1$ lying in a hyperplane. The authors define a set $B^*(Y)$ of units whose logarithmic embeddings lie within a box of side length $2Y$ . The product set of this set is small because multiplication of units corresponds to addition in the logarithmic lattice.
Set Definition: The target set $A$ is constructed as the product set $A = GP$, where $G = B^*(Y)$ (units) and $P = R + B^+(\epsilon R)$ (shifted algebraic integers). The parameters $R$ and $Y$ are chosen such that $R$ is sufficiently large relative to $Y$ (specifically $e^Y \leq \epsilon R$ ) to ensure the separation of the unit lattice and the additive structure.
Size Analysis:
- The size of $A$ is approximately $|G||P|$ .
- The product set $AA$ is bounded by analyzing $GG$ and $PP$. The smallness of $GG$ is derived from the lattice properties of the units.
- The shifted product set $(A+1)(A+1)$ is analyzed by rewriting elements as $g(p+1) = g(p + g^{-1})$ . This allows the authors to bound the set using the sumset $P + G$ . By choosing $R$ large enough, $P+G$ is shown to be contained within a set $P'$ whose size is comparable to $P$ .

Key Contributions and Results
The primary result is Theorem 1, which establishes the existence of an absolute constant $c > 0$ and arbitrarily large finite sets $A \subseteq \mathbb{R}$ such that:
$\max\{|AA|, |(A+1)(A+1)|\} \ll |A|^{2-c}$
This result demonstrates that the exponent in the inequality for the maximum of these two sets cannot be arbitrarily close to 2, thereby disproving the aforementioned folklore conjecture.

The paper further extends this result in Theorem 3 to "Many Translates." It shows that for any finite set of integers $S \subseteq \mathbb{Z}$ , there exists a constant $c > 0$ (dependent on $S$ ) and arbitrarily large sets $A$ such that:
$\max_{\lambda \in S} \{|(A+\lambda)(A+\lambda)|\} \ll |A|^{2-c}$
Additionally, the authors note that their construction implies that the set $A$ also satisfies $\max\{|A+A|, |AA|, |(A+1)(A+1)|\} \ll |A|^{2-c}$ and $|A(A+1)| \ll |A|^{2-c}$ .

Significance
The paper claims significance by providing a counterexample to a widely believed assumption in additive combinatorics: that additive shifts necessarily force a significant expansion in multiplicative structure. By adapting the BSZ construction, the authors show that it is possible to construct sets where both the product set and the shifted product set are significantly smaller than the trivial bound of $|A|^2$ . This work challenges the intuition that the exponent 2 is the natural limit for such sum-product type inequalities over the reals.

The authors explicitly state that the construction uses the same underlying set as the BSZ paper, but the specific application to the shifted product set $(A+1)(A+1)$ constitutes the novel contribution of this note. The work serves as a modification of existing arguments to refute a specific conjecture regarding the maximum of product and shifted product sets.