On the structure of approximate rings

Krzysztof Krupi\'nski; Simon Machado

arxiv: 2604.04581 · v1 · submitted 2026-04-06 · 🧮 math.RA · math.CO· math.LO

On the structure of approximate rings

Krzysztof Krupi\'nski , Simon Machado This is my paper

Pith reviewed 2026-05-10 19:50 UTC · model grok-4.3

classification 🧮 math.RA math.COmath.LO

keywords approximate subringsstructure theoremnilpotent quotientssum-product phenomenonGromov's theoremlocally compact modelsring theory

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The pith

Finite approximate subrings are controlled by nilpotent quotients that obstruct their growth under addition and multiplication.

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

The paper proves a structure theorem for finite approximate subrings in arbitrary rings. It identifies nilpotent quotients as the fundamental obstruction to simultaneous growth in addition and multiplication. This creates a uniform framework for the sum-product phenomenon that works in any ring. The result also provides a ring-theoretic version of Gromov's theorem on polynomial growth. The proof relies on the existence of definable locally compact models for these approximate sets.

Core claim

We prove that any finite approximate subring admits a definable locally compact model in which the obstruction to growth under both addition and multiplication is precisely the existence of nilpotent quotients. This theorem allows the development of sum-product results uniformly across all rings and yields as an application a counterpart to Gromov's theorem for rings of polynomial growth.

What carries the argument

Definable locally compact models for approximate subrings, which reduce the finite case to the analysis of quotients by nilpotent subgroups in a continuous setting.

If this is right

Approximate subrings without large nilpotent quotients must grow substantially when combining addition and multiplication.
Polynomial growth implies that the approximate subring is virtually nilpotent.
The structure extends to uniformly discrete approximate subrings in semi-simple real algebras, generalizing earlier results.
The modeling technique applies to approximate subrings beyond the finite setting.

Where Pith is reading between the lines

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

This framework could be used to investigate approximate subrings in non-associative algebras or other generalized structures.
Explicit quantitative bounds on growth might be derivable from the size of the nilpotent quotient.
The approach may connect to classification problems in additive combinatorics over general rings.

Load-bearing premise

That arbitrary approximate subrings possess definable locally compact models.

What would settle it

Constructing a finite approximate subring with slow growth in both operations but no significant nilpotent quotient in its model would disprove the identification of the obstruction.

read the original abstract

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate subrings. Our aim is to develop a general framework for the sum-product phenomenon that applies uniformly across arbitrary rings. The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication. Another application of the main structure theorem is a ring-theoretic counterpart of Gromov's theorem on groups of polynomial growth. The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24]. This existence theorem extends beyond the finite (and pseudofinite) setting. To illustrate the scope of the method, we also establish a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing a classical sum-product result of Meyer.

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

The paper gives a structure theorem for finite approximate subrings that flags nilpotent quotients as the main obstruction to joint additive-multiplicative growth, but the argument rests squarely on an earlier existence result for definable models.

read the letter

The paper's main contribution is a structure theorem for finite approximate subrings in arbitrary rings, where nilpotent quotients turn out to be the obstruction to growth under addition and multiplication. This sets up a uniform framework for sum-product phenomena and yields a ring version of Gromov's polynomial growth theorem. It does a good job extending previous work beyond specific cases like fields. By relying on definable locally compact models from the cited prior result, they reduce questions about approximate subrings to properties of locally compact rings, making the nilpotent part visible. The extra result on uniformly discrete approximate subrings in semi-simple real algebras generalizes Meyer's theorem in a natural way and shows the method's flexibility. The main soft spot is the dependence on that earlier existence theorem for the models. The paper takes it as given without reproving or detailing how every finite approximate subring satisfies the conditions, so the strength of the new theorem tracks the strength of the old one. If the prior work has any gaps in applicability, they would show up here too. The abstract gives no proof outline, which makes it tough to judge the extraction of the nilpotent quotient without the full text, but the citation is transparent and there's no sign of self-referential issues. This work is aimed at combinatorial algebraists and model theorists interested in algebraic structures with approximate operations. A reader who follows sum-product results or growth theorems in groups and rings will find the uniform statement useful, though they may need to consult the reference for the foundational step. It deserves serious refereeing because the claim is non-routine and the approach is logically connected to existing tools. I would recommend sending it out for peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The paper defines K-approximate subrings of an arbitrary ring and proves a structure theorem for finite approximate subrings. The main result identifies nilpotent quotients as the fundamental obstruction to simultaneous additive and multiplicative growth. The proof invokes the existence of definable locally compact models for arbitrary approximate subrings from the cited prior work [Kru24]. Applications include a ring-theoretic analogue of Gromov's theorem on groups of polynomial growth and a structure theorem for uniformly discrete approximate subrings of semi-simple real algebras, generalizing Meyer's classical sum-product result.

Significance. If the central claim holds, the work supplies a uniform model-theoretic framework for sum-product phenomena that applies across arbitrary rings rather than being restricted to fields or specific classes. The identification of nilpotent quotients as the obstruction and the Gromov-type application are potentially significant contributions. The extension to uniformly discrete approximate subrings in real algebras demonstrates broader scope beyond the finite case.

major comments (2)

[Abstract and §1 (Introduction)] The principal tool is the existence of definable locally compact models from [Kru24], invoked without reproof or explicit verification that every finite approximate subring satisfies the hypotheses of that theorem. This dependence is load-bearing for the structure theorem and the identification of nilpotent quotients as the obstruction, yet the manuscript supplies no independent check or derivation of how the model's topological/algebraic properties translate back to the original ring without additional unstated assumptions.
[Main structure theorem (likely §3 or §4)] The claim that nilpotent quotients constitute the fundamental obstruction to growth under both operations is derived directly from the properties of the definable model. Without a self-contained argument or explicit reduction showing that this obstruction follows from the input data alone (rather than from the external model), the central claim cannot be assessed for internal consistency from the given text.

minor comments (2)

[§1] The definition of approximate subring uses the phrasing 'covered by finitely many [resp. K] additive translates of X'; an early concrete example in a non-commutative ring would clarify the distinction between the finite and K-bounded cases.
[Abstract and references] The manuscript cites [Kru24] as the source of the key existence theorem; ensure that the statement of the main result explicitly delineates what is new in the present work versus direct application of the prior result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The work relies on the definable locally compact models from [Kru24] as the principal tool, which we view as applicable to finite approximate subrings by the general statement of that theorem. We agree that additional explicit verification and reduction steps would improve clarity and self-containedness, and we will incorporate revisions to address the concerns.

read point-by-point responses

Referee: [Abstract and §1 (Introduction)] The principal tool is the existence of definable locally compact models from [Kru24], invoked without reproof or explicit verification that every finite approximate subring satisfies the hypotheses of that theorem. This dependence is load-bearing for the structure theorem and the identification of nilpotent quotients as the obstruction, yet the manuscript supplies no independent check or derivation of how the model's topological/algebraic properties translate back to the original ring without additional unstated assumptions.

Authors: We agree that an explicit verification is warranted for accessibility. In the revised version, we will add a short paragraph (or subsection) in §1 or §2 confirming that finite K-approximate subrings satisfy the hypotheses of the existence theorem in [Kru24]: they are definable in the ring language, additively symmetric, and satisfy the finite covering conditions for X·X ∪ (X+X). This is a direct special case of the general result for arbitrary approximate subrings. We will also include a brief derivation showing the correspondence: the definable embedding into the locally compact model preserves the approximate operations, so nilpotent quotients in the model correspond to obstructions in the original ring's additive and multiplicative growth via the quotient maps induced by the model. This addresses the translation without unstated assumptions. revision: yes
Referee: [Main structure theorem (likely §3 or §4)] The claim that nilpotent quotients constitute the fundamental obstruction to growth under both operations is derived directly from the properties of the definable model. Without a self-contained argument or explicit reduction showing that this obstruction follows from the input data alone (rather than from the external model), the central claim cannot be assessed for internal consistency from the given text.

Authors: We acknowledge that the current presentation could be more explicit about the reduction. In the revised manuscript, we will expand the proof of the main structure theorem (in the relevant section) with a numbered outline of the argument: (i) invoke the model theorem to obtain a definable locally compact ring containing a model of the approximate subring; (ii) use the model's topological and algebraic properties to derive that non-nilpotent quotients would imply exponential growth in either addition or multiplication (contradicting the approximate subring assumption); (iii) project the nilpotent quotient back to the original ring via the definable map, showing it obstructs simultaneous growth in the input data. This step-by-step reduction will be self-contained in the text while citing [Kru24] only for the model existence, allowing internal consistency to be assessed directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives its structure theorem for finite approximate subrings and identifies nilpotent quotients as the growth obstruction by applying the existence of definable locally compact models as a principal tool from the cited prior work [Kru24]. This constitutes standard dependence on an external theorem rather than any self-definitional loop, fitted input renamed as prediction, or reduction of the new claims to the paper's own inputs by construction. The applications to a ring-theoretic Gromov theorem and uniformly discrete approximate subrings in semi-simple algebras add independent content. No equations or steps in the provided text exhibit the specific reductions required for flagging circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on a single external existence theorem for definable models; no free parameters or new invented entities are introduced in the abstract.

axioms (1)

domain assumption Existence of definable locally compact models for arbitrary approximate subrings
Cited as the principal tool from prior work [Kru24].

pith-pipeline@v0.9.0 · 5476 in / 1241 out tokens · 65811 ms · 2026-05-10T19:50:03.266382+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The main result identifies nilpotent quotients as the fundamental obstruction to growth under both addition and multiplication... The principal tool in the proof is the existence of definable locally compact models for arbitrary approximate subrings from [Kru24].
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

||x+y||_Y ≤ 4 max(||x||_Y, ||y||_Y) ≤ 4(||x||_Y + ||y||_Y) and ||xy||_Y ≤ 2||x||_Y · ||y||_Y

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Differential operators onG/Uand the Gelfand-Graev action

[BH18] Michael Björklund and Tobias Hartnick. “Approximate lattices”. In:Duke Mathe- matical Journal167.15 (2018), pp. 719–743 (cit. on p. 32). [Bou05a] J. Bourgain. “Mordell’s exponential sum estimate revisited”. English. In:J. Am. Math. Soc.18.2 (2005), pp. 477–499.doi:10.1090/S0894- 0347- 05- 00476- 5 (cit. on p. 2). 46 REFERENCES [BKT04] J. Bourgain, ...

work page doi:10.1090/s0894- 2018
[2]

A Free Probability Analogue of the Wasserstein Metric on the Trace- State Space

arXiv:2011.12009v1(cit. on pp. 4, 5, 31, 40). [HKP22] EhudHrushovski,KrzysztofKrupiński,andAnandPillay.“Amenability,connected components, and definable actions”. In:Sel. Math. New Ser.28, 16 (2022) (cit. on pp. 5, 41). [JT35] Nathan Jacobson and Olga Taussky. “Locally compact rings”. In:Proceedings of the National Academy of Sciences21.2 (1935), pp. 106–1...

work page doi:10.1007/s00039- 2011
[3]

Berlin etc.: Springer-Verlag, 1991 (cit

Folge. Berlin etc.: Springer-Verlag, 1991 (cit. on p. 34). [Mey72a] Yves Meyer.Algebraic numbers and harmonic analysis. Vol

work page 1991
[4]

Elsevier, 1972 (cit. on pp. 4, 7, 30, 31). [Mey72b] Yves Meyer.Algebraic numbers and harmonic analysis. Vol

work page 1972
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Grundlehren Math. Wiss. Berlin: Springer, 1999 (cit. on pp. 30, 34). [Pie82] Richard S. Pierce.Associative algebras. English. Vol

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Sum-product phenomena:P-adic case

Grad. Texts Math. Springer, Cham, 1982 (cit. on pp. 31, 34, 35, 38, 45). [SG20] Alireza Salehi Golsefidy. “Sum-product phenomena:P-adic case”. English. In:J. Anal. Math.142.2 (2020), pp. 349–419.doi:10.1007/s11854-020-0139-y(cit. on p. 3). [Sch73] Jean-Pierre Schreiber. “Approximations diophantiennes et problemes additifs dans les groupes abéliens localem...

work page doi:10.1007/s11854-020-0139-y(cit 1982
[7]

Approximate groups [according to Hrushovski and Breuillard, Green, Tao]

Cambridge Univer- sity Press, 2012 (cit. on p. 8). [VDD15] Lou Van Den Dries. “Approximate groups [according to Hrushovski and Breuillard, Green, Tao]”. In:Astérisque. Société mathématique de France, 2015, pp. 79–113 (cit. on pp. 6, 10, 13, 17, 21, 26, 34, 38). [Yam53] Hidehiko Yamabe. “On the conjecture of Iwasawa and Gleason”. In:Annals of Mathematics58...

work page 2012

[1] [1]

Differential operators onG/Uand the Gelfand-Graev action

[BH18] Michael Björklund and Tobias Hartnick. “Approximate lattices”. In:Duke Mathe- matical Journal167.15 (2018), pp. 719–743 (cit. on p. 32). [Bou05a] J. Bourgain. “Mordell’s exponential sum estimate revisited”. English. In:J. Am. Math. Soc.18.2 (2005), pp. 477–499.doi:10.1090/S0894- 0347- 05- 00476- 5 (cit. on p. 2). 46 REFERENCES [BKT04] J. Bourgain, ...

work page doi:10.1090/s0894- 2018

[2] [2]

A Free Probability Analogue of the Wasserstein Metric on the Trace- State Space

arXiv:2011.12009v1(cit. on pp. 4, 5, 31, 40). [HKP22] EhudHrushovski,KrzysztofKrupiński,andAnandPillay.“Amenability,connected components, and definable actions”. In:Sel. Math. New Ser.28, 16 (2022) (cit. on pp. 5, 41). [JT35] Nathan Jacobson and Olga Taussky. “Locally compact rings”. In:Proceedings of the National Academy of Sciences21.2 (1935), pp. 106–1...

work page doi:10.1007/s00039- 2011

[3] [3]

Berlin etc.: Springer-Verlag, 1991 (cit

Folge. Berlin etc.: Springer-Verlag, 1991 (cit. on p. 34). [Mey72a] Yves Meyer.Algebraic numbers and harmonic analysis. Vol

work page 1991

[4] [4]

Elsevier, 1972 (cit. on pp. 4, 7, 30, 31). [Mey72b] Yves Meyer.Algebraic numbers and harmonic analysis. Vol

work page 1972

[5] [5]

Grundlehren Math. Wiss. Berlin: Springer, 1999 (cit. on pp. 30, 34). [Pie82] Richard S. Pierce.Associative algebras. English. Vol

work page 1999

[6] [6]

Sum-product phenomena:P-adic case

Grad. Texts Math. Springer, Cham, 1982 (cit. on pp. 31, 34, 35, 38, 45). [SG20] Alireza Salehi Golsefidy. “Sum-product phenomena:P-adic case”. English. In:J. Anal. Math.142.2 (2020), pp. 349–419.doi:10.1007/s11854-020-0139-y(cit. on p. 3). [Sch73] Jean-Pierre Schreiber. “Approximations diophantiennes et problemes additifs dans les groupes abéliens localem...

work page doi:10.1007/s11854-020-0139-y(cit 1982

[7] [7]

Approximate groups [according to Hrushovski and Breuillard, Green, Tao]

Cambridge Univer- sity Press, 2012 (cit. on p. 8). [VDD15] Lou Van Den Dries. “Approximate groups [according to Hrushovski and Breuillard, Green, Tao]”. In:Astérisque. Société mathématique de France, 2015, pp. 79–113 (cit. on pp. 6, 10, 13, 17, 21, 26, 34, 38). [Yam53] Hidehiko Yamabe. “On the conjecture of Iwasawa and Gleason”. In:Annals of Mathematics58...

work page 2012