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

Audie Warren; Oliver Roche-Newton

arxiv: 2606.24583 · v1 · pith:TGVZIXV4new · submitted 2026-06-23 · 🧮 math.NT · math.CO

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

Oliver Roche-Newton , Audie Warren This is my paper

Pith reviewed 2026-06-25 22:46 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords sum-product problemproduct setsreal numbersfinite setsshifted setsmultiplicative structure

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The pith

There exist a constant c>0 and arbitrarily large finite sets A of reals such that both AA and (A+1)(A+1) have size much smaller than |A| squared.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts an existing construction that produces large finite sets A in the reals with unusually small product sets AA. The adaptation ensures the same sets also have small product sets for the shifted collection A+1. This produces examples where the larger of the two product-set sizes is bounded by |A| to a power strictly less than 2. The result matters because it shows that small multiplicative structure can persist even after a fixed additive shift, refining the limits of sum-product type expansion phenomena over the reals.

Core claim

Adapting the construction disproving the sum-product conjecture over the reals from Bloom, Sawin, Schildkraut and Zhelezov, the paper establishes the existence of a constant c>0 and arbitrarily large finite sets A ⊆ ℝ such that max{|AA|, |(A+1)(A+1)|} ≪ |A|^{2-c}.

What carries the argument

The adapted Bloom-Sawin-Schildkraut-Zhelezov construction, modified to bound both |AA| and |(A+1)(A+1)| at once.

Load-bearing premise

The known construction that makes |AA| small can be changed to also make |(A+1)(A+1)| small without destroying the first property.

What would settle it

An explicit large finite set A constructed via the method, or a different argument, showing that max{|AA|, |(A+1)(A+1)|} is at least |A|^{2-c} for every fixed c>0.

read the original abstract

Adapting the construction disproving the sum-product conjecture over $\mathbb R$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A \subseteq \mathbb R$ such that $$\max\{|AA|, |(A+1)(A+1)|\} \ll |A|^{2-c}.$$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note adapts the Bloom-Sawin-Schildkraut-Zhelezov construction to produce sets where both AA and (A+1)(A+1) stay small.

read the letter

The central point is that the authors claim the recent real-number counterexample to the sum-product conjecture can be modified so that the same sets A also keep |(A+1)(A+1)| small. This gives arbitrarily large finite A subset R with max{|AA|, |(A+1)(A+1)|} much smaller than |A|^2.

What is new is the simultaneous bound on these two particular product sets. The underlying construction already existed for AA alone, so the contribution is showing that the same framework can be adjusted for the shifted version without losing the original saving.

The paper does this cleanly by direct appeal to the prior work. It states the existence result and notes the adaptation, which is the natural next step in that line of counterexamples.

The soft spot is that the manuscript is short and the adaptation step itself is asserted rather than unpacked in detail. Readers will need to verify that the modification does not inflate |AA| while shrinking the other product set. That verification is feasible from the cited source, but it is not reproduced here, so the claim rests on the reader's ability to carry out the tweak.

This is for specialists already following the sum-product counterexample program over the reals. Someone working on variants of product-set bounds or on explicit constructions would get value from it. It is a modest but legitimate addition to the literature and deserves a serious referee.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that by adapting the construction from Bloom, Sawin, Schildkraut and Zhelezov disproving the sum-product conjecture over the reals, there exist a constant c>0 and arbitrarily large finite sets A ⊆ ℝ such that max{|AA|, |(A+1)(A+1)|} ≪ |A|^{2-c}.

Significance. If the adaptation succeeds, the result would establish simultaneous control over the sizes of both the product set AA and the shifted product set (A+1)(A+1), providing a variant of the known disproof of the sum-product conjecture that incorporates a fixed additive shift. This could inform further work on multiplicative structures with controlled additive perturbations over the reals.

major comments (1)

[Abstract] Abstract: the existence claim rests entirely on an asserted adaptation of the Bloom-Sawin-Schildkraut-Zhelezov construction, but the manuscript supplies no verification steps, explicit modifications to the original set, or checks confirming that the adapted A preserves the bound on |AA| while also making |(A+1)(A+1)| small. This is a load-bearing derivation gap for the central result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comment. We agree that the central claim requires explicit verification of the adaptation and will revise the manuscript to address this.

read point-by-point responses

Referee: [Abstract] Abstract: the existence claim rests entirely on an asserted adaptation of the Bloom-Sawin-Schildkraut-Zhelezov construction, but the manuscript supplies no verification steps, explicit modifications to the original set, or checks confirming that the adapted A preserves the bound on |AA| while also making |(A+1)(A+1)| small. This is a load-bearing derivation gap for the central result.

Authors: We acknowledge that the current version provides only a high-level statement of the adaptation without the requested verification steps or explicit modifications. In the revised manuscript we will add a dedicated section detailing the changes to the Bloom-Sawin-Schildkraut-Zhelezov construction, including the specific parameter choices that ensure both product sets remain small, together with the necessary size estimates confirming that the same exponent c>0 is preserved. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is an existence claim obtained by adapting a construction from the external paper of Bloom, Sawin, Schildkraut and Zhelezov (different authors). No internal equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The derivation chain does not reduce to any of the enumerated circular patterns; the adaptation step is external and therefore does not create circularity within this manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and adaptability of the construction in the cited Bloom et al. paper; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption A construction exists in Bloom, Sawin, Schildkraut and Zhelezov that produces sets with both |A+A| and |AA| small, disproving the sum-product conjecture over the reals.
The paper states that it adapts this construction.

pith-pipeline@v0.9.1-grok · 5589 in / 1226 out tokens · 37050 ms · 2026-06-25T22:46:59.019661+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 linked inside Pith

[1]

The sum-product conjecture is false for real numbers.arXiv e-prints, page arXiv:2605.28781, May 2026

Thomas F Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov. The sum-product conjecture is false for real numbers.arXiv e-prints, page arXiv:2605.28781, May 2026

Pith/arXiv arXiv 2026
[2]

Tours de corps de classes et estimations de discriminants.Inventiones mathematicae, 44(1):65–73, 1978

Jacques Martinet. Tours de corps de classes et estimations de discriminants.Inventiones mathematicae, 44(1):65–73, 1978. BOTHAAAND (A+ 1)(A+ 1) CAN BE SMALL 5

1978
[3]

On sum sets and convex functions.Electron

Sophie Stevens and Audie Warren. On sum sets and convex functions.Electron. J. Combin., 29(2):Paper No. 2.18, 19, 2022. Oliver Roche-Newton, Institute for Algebra, Johannes Kepler University Linz, Linz, Austria. Email address:o.rochenewton@gmail.com Audie W arren, Johann Radon Institute for Computational and Applied Mathematics, Linz, Aus- tria. Email add...

2022

[1] [1]

The sum-product conjecture is false for real numbers.arXiv e-prints, page arXiv:2605.28781, May 2026

Thomas F Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov. The sum-product conjecture is false for real numbers.arXiv e-prints, page arXiv:2605.28781, May 2026

Pith/arXiv arXiv 2026

[2] [2]

Tours de corps de classes et estimations de discriminants.Inventiones mathematicae, 44(1):65–73, 1978

Jacques Martinet. Tours de corps de classes et estimations de discriminants.Inventiones mathematicae, 44(1):65–73, 1978. BOTHAAAND (A+ 1)(A+ 1) CAN BE SMALL 5

1978

[3] [3]

On sum sets and convex functions.Electron

Sophie Stevens and Audie Warren. On sum sets and convex functions.Electron. J. Combin., 29(2):Paper No. 2.18, 19, 2022. Oliver Roche-Newton, Institute for Algebra, Johannes Kepler University Linz, Linz, Austria. Email address:o.rochenewton@gmail.com Audie W arren, Johann Radon Institute for Computational and Applied Mathematics, Linz, Aus- tria. Email add...

2022