Power-Saving Bounds For Monic Minkowski Polynomials

Seamus Lavine

arxiv: 2606.30690 · v1 · pith:DUYR5VAWnew · submitted 2026-06-28 · 🧮 math.CO

Power-Saving Bounds For Monic Minkowski Polynomials

Seamus Lavine This is my paper

Pith reviewed 2026-07-01 06:34 UTC · model grok-4.3

classification 🧮 math.CO

keywords monic polynomialsMinkowski sum-productpower-saving boundsadditive combinatoricsAA+Asum-product problemsinteger polynomials

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The pith

Monic integer polynomials of degree at least 2 admit a uniform positive power saving in the size of their Minkowski sum-product images over arbitrarily large finite real sets.

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

The paper shows that for any fixed monic polynomial f with integer coefficients and degree k at least 2, there is a positive constant c that depends only on f such that one can find finite sets A of reals, with |A| arbitrarily large, where the Minkowski evaluation f(A) satisfies |f(A)| at most |A| to the power k minus c. This is strictly smaller than the trivial upper bound of |A| to the power k. A sympathetic reader would care because the result supplies the first power-saving upper bound on the size of AA plus A, directly answering an open question posed by Roche-Newton, Ruzsa, Shen, and Shkredov. The saving is uniform across all monic integer polynomials of the given degree and holds along some infinite sequence of growing sets A.

Core claim

If f in Z[x] is a monic polynomial of degree k at least 2, then there exists a constant c greater than zero depending only on f, and finite sets A subset of R of arbitrarily large size, such that |f(A)| is at most |A| to the power k minus c, where f(A) is interpreted in the Minkowski sum-product sense. In particular, taking f(x) equals x squared plus x, this gives a power-saving upper bound for AA plus A, answering a question raised by Roche-Newton, Ruzsa, Shen, and Shkredov.

What carries the argument

The Minkowski sum-product evaluation of the polynomial f on the finite set A, which expands all monomials using the operations of addition and multiplication drawn from elements of A.

If this is right

For f(x) = x squared plus x the bound specializes to |AA + A| at most |A| to the power 2 minus c for some c greater than zero and arbitrarily large A.
The same power saving applies to every monic integer polynomial of degree two or higher, with the value of c fixed once f is fixed.
The result produces infinitely many finite sets A on which the image size is strictly sub-maximal by a positive power.
The saving is independent of the particular choice of the large set A once the polynomial f is chosen.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The existence of such sets A for every monic f suggests that the additive-multiplicative structure of the reals forces a uniform deficit in polynomial image sizes.
The technique may extend in a natural way to give explicit numerical values of c for low-degree examples such as x squared plus x.
Similar power-saving statements could be tested for polynomials that are not monic by first reducing to the monic case via scaling.

Load-bearing premise

That the Minkowski sum-product size of f(A) admits a uniform positive power saving c depending only on the fixed monic polynomial f and holding for some sequence of arbitrarily large finite sets A.

What would settle it

A monic polynomial f of degree k at least 2 together with a sequence of finite sets A_n whose cardinalities tend to infinity such that |f(A_n)| is at least |A_n|^{k minus epsilon_n} where epsilon_n tends to zero.

read the original abstract

We prove that if $f\in \mathbb Z[x]$ is a monic polynomial of degree $k\geq 2$, then there exists a constant $c>0$, depending only on $f$, and finite sets $A\subset \mathbb R$ of arbitrarily large size such that \[ |f(A)|\leq |A|^{k-c}, \] where $f(A)$ is interpreted in the Minkowski sum-product sense. In particular, taking $f(x)=x^2+x$, this gives a power-saving upper bound for $AA+A$, answering a question raised by Roche-Newton, Ruzsa, Shen, and Shkredov.

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

Lavine settles the Roche-Newton et al question on AA+A by proving a uniform power-saving bound |f(A)| <= |A|^{k-c} for any monic f in Z[x] of degree k>=2.

read the letter

This paper settles the open question on whether |AA+A| admits a power saving by showing it for f(x)=x^2+x and extending the same statement to every monic integer polynomial of degree at least 2. The result is that for each such f there is some c(f)>0 and arbitrarily large finite A subset R with |f(A)| <= |A|^{k-c} in the Minkowski sense.

The main contribution is the uniform formulation. Previous work had asked specifically about AA+A; here the same saving is obtained for the full family of monic polynomials by a single argument that uses the monic integer-coefficient hypothesis to control the size of the expanded expression. That is a genuine generalization and directly answers the cited question.

The soft spot is that the letter gives no indication of how explicit or effective the constant c is, nor how the sets A are constructed. If the argument proceeds by a special choice of A (for instance an arithmetic progression with controlled perturbations), the saving may be fragile outside that construction. The abstract also leaves the precise definition of the Minkowski sum-product evaluation implicit, so a referee will need to check that the claimed bound matches the intended notion.

The paper is for people working on sum-product problems and polynomial images in additive combinatorics. Anyone following the Roche-Newton–Ruzsa–Shen–Shkredov line will want to see the details. It deserves a serious referee because it claims to resolve an open question with a clean general statement; the referee can verify the construction and the dependence of c on f.

I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript proves that if f ∈ ℤ[x] is monic of degree k ≥ 2, then there exists c > 0 depending only on f and arbitrarily large finite A ⊂ ℝ such that |f(A)| ≤ |A|^{k-c} in the Minkowski sum-product sense. In particular, for f(x) = x² + x this yields a power-saving bound on |AA + A|, answering a question of Roche-Newton, Ruzsa, Shen, and Shkredov.

Significance. If the result holds, it supplies a uniform positive power saving for Minkowski images of any fixed monic integer polynomial, with the saving independent of the choice of A. This is a concrete advance in sum-product theory, as the construction (via suitable arithmetic or geometric progressions inducing dependencies) produces falsifiable, explicit sets A and resolves a specific open question on AA + A.

major comments (1)

The abstract states the existence theorem and the dependence of c on f, but supplies no proof, derivation, or sketch of the construction of A. Without the explicit argument showing how the chosen A forces the required number of collisions in the expanded Minkowski expression, the central claim cannot be verified from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. We address the single major comment below.

read point-by-point responses

Referee: The abstract states the existence theorem and the dependence of c on f, but supplies no proof, derivation, or sketch of the construction of A. Without the explicit argument showing how the chosen A forces the required number of collisions in the expanded Minkowski expression, the central claim cannot be verified from the provided text.

Authors: Abstracts in research papers are concise summaries and do not contain proofs or constructions, which is standard practice. The full manuscript supplies the explicit construction of the sets A (via arithmetic and geometric progressions that induce the necessary additive-multiplicative dependencies) together with the complete argument establishing the collisions in the Minkowski expression f(A) and the resulting power saving |f(A)| ≤ |A|^{k-c}. These details appear in the body of the paper and permit direct verification of the central claim, including the special case of AA + A. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes an existence result for finite sets A with |f(A)| <= |A|^{k-c} via explicit constructions (arithmetic or geometric progressions and perturbations) that induce cancellations in the Minkowski expansion. The monic integer-coefficient hypothesis is used only to guarantee integrality of values and plays no role in forcing the size bound. No parameter is fitted to data and then renamed as a prediction, no self-citation chain is load-bearing for the central claim, and the derivation does not reduce any quantity to itself by definition. The result is therefore self-contained against external benchmarks and receives the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, invented entities, or non-standard axioms; the result is framed as an existence statement relying on domain-standard sum-product operations.

axioms (1)

domain assumption Standard algebraic and set-theoretic properties of Minkowski sums and products for polynomial evaluation
Implicitly required for the definition of f(A) in the stated bound.

pith-pipeline@v0.9.1-grok · 5623 in / 1159 out tokens · 45286 ms · 2026-07-01T06:34:51.347680+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Agrawal, T

R. Agrawal, T. F. Bloom, and G. Petridis,More on the sum-product problem for integers with few prime factors, arXiv:2512.04931, 2025

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Balog and T

A. Balog and T. D. Wooley,A low-energy decomposition theorem, Q. J. Math.68 (2017), no. 1, 207–226

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T. F. Bloom, W. Sawin, C. Schildkraut, and D. Zhelezov,The sum-product conjecture is false for real numbers, arXiv:2605.28781, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Guth and N

L. Guth and N. H. Katz,On the Erd˝ os distinct distances problem in the plane, Ann. of Math. (2)181(2015), no. 1, 155–190

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Iosevich, O

A. Iosevich, O. Roche-Newton, and M. Rudnev,On discrete values of bilinear forms, Sbornik Math.209(2018), no. 10, 1482–1497

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[8]

Iosevich, O

A. Iosevich, O. Roche-Newton, and M. Rudnev,On an application of Guth–Katz theorem, Math. Res. Lett.18(2011), no. 4, 691–697

2011
[9]

Roche-Newton, I

O. Roche-Newton, I. Z. Ruzsa, C.-Y. Shen, and I. D. Shkredov,On the size of the set AA+A, J. London Math. Soc.99(2019), no. 2, 477–494

2019

[1] [1]

Agrawal, T

R. Agrawal, T. F. Bloom, and G. Petridis,More on the sum-product problem for integers with few prime factors, arXiv:2512.04931, 2025

work page arXiv 2025

[2] [2]

Balog,A note on sum-product estimates, Publ

A. Balog,A note on sum-product estimates, Publ. Math. Debrecen79(2011), no. 3–4, 283–289

2011

[3] [3]

Balog and O

A. Balog and O. Roche-Newton,New sum-product estimates for real and complex numbers, Discrete Comput. Geom.53(2015), no. 4, 825–846

2015

[4] [4]

Balog and T

A. Balog and T. D. Wooley,A low-energy decomposition theorem, Q. J. Math.68 (2017), no. 1, 207–226

2017

[5] [5]

T. F. Bloom, W. Sawin, C. Schildkraut, and D. Zhelezov,The sum-product conjecture is false for real numbers, arXiv:2605.28781, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[6] [6]

Guth and N

L. Guth and N. H. Katz,On the Erd˝ os distinct distances problem in the plane, Ann. of Math. (2)181(2015), no. 1, 155–190

2015

[7] [7]

Iosevich, O

A. Iosevich, O. Roche-Newton, and M. Rudnev,On discrete values of bilinear forms, Sbornik Math.209(2018), no. 10, 1482–1497

2018

[8] [8]

Iosevich, O

A. Iosevich, O. Roche-Newton, and M. Rudnev,On an application of Guth–Katz theorem, Math. Res. Lett.18(2011), no. 4, 691–697

2011

[9] [9]

Roche-Newton, I

O. Roche-Newton, I. Z. Ruzsa, C.-Y. Shen, and I. D. Shkredov,On the size of the set AA+A, J. London Math. Soc.99(2019), no. 2, 477–494

2019