The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

Bal\'azs Szegedy; Diego Gonz\'alez-S\'anchez; Pablo Candela

arxiv: 2601.08810 · v2 · pith:BJOCV2I5new · submitted 2026-01-13 · 🧮 math.GR · math.CO

The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

Pablo Candela , Diego Gonz\'alez-S\'anchez , Bal\'azs Szegedy This is my paper

Pith reviewed 2026-05-16 14:28 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords Jamneshan-Tao conjectureGowers normsinverse theoremsnilsequencesfinite abelian groupsbounded rankadditive combinatorics

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The pith

The Jamneshan-Tao conjecture holds for all finite abelian groups generated by at most R elements.

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

The paper establishes the Jamneshan-Tao conjecture in the case of finite abelian groups of rank bounded by a fixed R. It does so by proving an inverse theorem: any 1-bounded function on such a group whose Gowers norm is non-trivial must correlate with a nilsequence whose complexity depends only on R and the size of the norm. This restriction to bounded rank lets the argument control the nilsequences that appear. Readers would care because the conjecture describes the structural consequences of large Gowers norms, which govern the presence of arithmetic configurations in additive groups.

Core claim

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer R by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.

What carries the argument

An inverse theorem that converts a non-trivial Gowers norm into correlation with a nilsequence of complexity controlled by the rank bound R.

If this is right

The conjecture is settled for every finite abelian group of rank ≤ R.
High Gowers-norm functions on these groups must be structured rather than random-like.
Higher-order Fourier analysis on bounded-rank groups reduces to correlation with nilsequences.
Results about arithmetic progressions or other configurations follow directly in the bounded-rank setting.

Where Pith is reading between the lines

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

The same conclusion may fail when rank grows with the group order, requiring new ideas to bound nilsequence complexity.
Small explicit checks on groups such as (Z/pZ)^k for fixed k could confirm the correlation statement numerically.
The result suggests that uniformity norms detect nilsequence structure once generation rank is fixed.

Load-bearing premise

The group is generated by at most R elements, which keeps the complexity of the nilsequences that arise under control.

What would settle it

A 1-bounded function on some group of rank at most R that has positive Gowers norm yet correlates with no nilsequence of complexity bounded in terms of R and the norm value.

read the original abstract

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer $R$ (i.e. finite abelian groups generated by at most $R$ elements), by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.

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 paper settles the Jamneshan-Tao conjecture for finite abelian groups of rank at most fixed R by proving a direct inverse theorem for Gowers norms.

read the letter

The main point is that Candela, González-Sánchez and Szegedy confirm the Jamneshan-Tao conjecture when the finite abelian group is generated by at most R elements. They prove that any 1-bounded function with non-trivial Gowers norm on such a group correlates with a nilsequence whose complexity depends only on R and the usual parameters. This is new because the unrestricted conjecture remains open and no prior work had handled even this bounded-rank case cleanly. The rank restriction is used exactly to keep nilsequence complexity finite and independent of group order, which is a practical way to make the inverse theorem usable for density-increment arguments on these groups. The statement itself is direct: no reduction to fitted quantities or self-referential definitions appears in the claim. The approach builds on standard nilsequence machinery without obvious reinvention. The soft spot is the lack of visible detail on error terms and complexity bounds in the abstract alone. One would need to check whether the quantitative dependence on R is reasonable and whether the approximation steps hold without hidden blow-up. Nothing in the stated claim suggests circularity or inconsistency, but the full derivation would need verification on those points. This work is aimed at people already working in higher-order Fourier analysis and additive combinatorics on abelian groups. Readers who follow inverse theorems or removal lemmas on restricted groups will find a usable statement here. It deserves serious peer review because the result is specific, the restriction is natural, and confirming even this case of the conjecture is worth referee time. I would send it out.

Referee Report

0 major / 1 minor

Summary. The manuscript confirms the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer R by establishing an inverse theorem: any 1-bounded function on such a group whose Gowers norm is non-trivial must correlate non-trivially with a nilsequence whose complexity is bounded in terms of R alone.

Significance. If correct, the result supplies a clean partial resolution of the conjecture by exploiting the bounded-rank hypothesis to keep nilsequence parameters finite and independent of group order. This supplies a concrete advance in the inverse theory of Gowers norms for abelian groups and furnishes a template that may be useful for the unbounded-rank case.

minor comments (1)

The dependence of the nilsequence complexity bound on the rank parameter R is stated only qualitatively in the abstract; an explicit functional dependence would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main result: an inverse theorem establishing the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed R, with nilsequence complexity depending only on R.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an inverse theorem for 1-bounded functions with non-trivial Gowers norm on finite abelian groups of rank at most R, showing correlation with nilsequences of bounded complexity. This directly confirms the Jamneshan-Tao conjecture under the bounded-rank restriction, which serves only to keep nilsequence parameters finite and independent of group order. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the argument relies on standard inverse theorem techniques adapted to the rank bound without renaming known results or smuggling ansatzes. The central claim has independent mathematical content and is not forced by prior self-referential inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the established theory of Gowers norms and nilsequences developed in prior work on higher-order Fourier analysis; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of Gowers norms and the correspondence between large Gowers norms and nilsequences hold on finite abelian groups
The inverse theorem invokes the existing framework of Gowers uniformity and nilsequence approximation without re-deriving it.

pith-pipeline@v0.9.0 · 5361 in / 1189 out tokens · 25436 ms · 2026-05-16T14:28:21.819539+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 2 internal anchors

[1]

Bergelson, T

V . Bergelson, T. Tao and T. Ziegler,An inverse theorem for the uniformity seminorms associated with the action ofF ∞ p , Geom. Funct. Anal.19(2010), no. 6, 1539–1596. 1

work page 2010
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Antol ´ın Camarena, B

O. Antol ´ın Camarena, B. Szegedy,Nilspaces, nilmanifolds and their morphisms, preprint (2010),http: //arxiv.org/abs/1009.38252

work page arXiv 2010
[3]

Candela,Notes on nilspaces: algebraic aspects, Discrete Anal., 2017, Paper No

P. Candela,Notes on nilspaces: algebraic aspects, Discrete Anal., 2017, Paper No. 15, 59 pp. 3, 4, 5, 6, 9, 10, 17

work page 2017
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Candela,Notes on compact nilspaces, Discrete Anal., 2017, Paper No

P. Candela,Notes on compact nilspaces, Discrete Anal., 2017, Paper No. 16, 57pp. 2, 4, 5, 6, 8, 9, 10, 11, 12

work page 2017
[5]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyOn nilspace systems and their morphisms, Ergodic Theory Dynam. Systems40(2020), no. 11, 3015–3029. 6

work page 2020
[6]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. Szegedy,On higher-order Fourier analysis in characteristicp, Ergodic Theory Dynam. Systems43(2023), no. 12, 3971-4040. 12

work page 2023
[7]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyFree nilspaces, double coset nilspaces, and Gowers norms, preprint (2025),https://arxiv.org/abs/2305.11233. 17, 18, 22

work page arXiv 2025
[8]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyOn the inverse theorem in abelian groups of bounded torsion, preprint (2023),https://arxiv.org/abs/2311.13899. 15

work page arXiv 2023
[9]

Candela, D

P. Candela, D. Gonz´alez-S´anchez, B. SzegedyAn inverse theorem for all finite abelian groups via nilmanifolds, preprint (2025),https://arxiv.org/abs/2512.17468. 1

work page arXiv 2025
[10]

Candela, B

P. Candela, B. Szegedy,Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds, J. Reine Angew. Math.789(2022), 1–42. 2, 4, 6, 8, 10, 11, 13

work page 2022
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W. T. Gowers,A new proof of Szemer ´edi’s theorem, Geom. Funct. Anal.11(2001), no. 3, 465–588

work page 2001
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W. T. Gowers, L. Mili´cevi´c,A quantitative inverse theorem for theU 4 norm over finite fields, preprint (2017), https://arxiv.org/abs/1712.00241. 1

work page internal anchor Pith review Pith/arXiv arXiv 2017
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Green, T

B. Green, T. Tao,An inverse theorem for the GowersU 3-norm, Proc. Edinburgh Math. Soc. (1)51(2008), 73–153. 1

work page 2008
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Green, T

B. Green, T. Tao,The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math.175(2012), 465–540. 22

work page 2012
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Green, T

B. Green, T. Tao, T. Ziegler,An inverse theorem for the GowersU 4-norm, Glasgow Math. J.53(2011), 1–50. 1 26 PABLO CANDELA, DIEGO GONZ ´ALEZ-S ´ANCHEZ, AND BAL ´AZS SZEGEDY

work page 2011
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Green, T

B. Green, T. Tao, T. Ziegler,An inverse theorem for the GowersU s+1[N]-norm, Ann. of Math. (2)176 (2012), no. 2, 1231–1372. 1, 16, 18, 21, 22

work page 2012
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K. H. Hofmann, S. A. Morris,The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie al- gebras, pro-Lie groups, and connected locally compact groups. EMS Tracts Math., 2. European Mathematical Society (EMS), Z¨urich, 2007. 17

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B. Host, B. Kra,Nilpotent structures in ergodic theory, Math. Surveys Monogr., 236. American Mathematical Society, Providence, RI, 2018. 16, 19

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Jamneshan, O

A. Jamneshan, O. Shalom, T. Tao,Polynomial towers and inverse Gowers theory for bounded-exponent groups, preprint (2026),https://arxiv.org/abs/2601.00961. 1

work page arXiv 2026
[20]

Jamneshan, T

A. Jamneshan, T. Tao,The inverse theorem for theU 3 Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches, Discrete Anal., 2023:11, 48 pp. 1, 4

work page 2023
[21]

J. Leng, A. Sah, M. Sawhney,Quasipolynomial bounds on the inverse theorem for the GowersU s+1[N]- norm, prerpint (2024),https://arxiv.org/abs/2402.17994. 1

work page arXiv 2024
[22]

Manners,Quantitative bounds in the inverse theorem for the GowersU s+1-norms over cyclic groups preprint (2018),http://www.arxiv.org/abs/1811.00718

F. Manners,Quantitative bounds in the inverse theorem for the GowersU s+1-norms over cyclic groups preprint (2018),http://www.arxiv.org/abs/1811.00718. 1

work page arXiv 2018
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Periodic nilsequences and inverse theorems on cyclic groups

F. Manners,Periodic nilsequences and inverse theorems on cyclic groupspreprint (2014),http://www. arxiv.org/abs/1404.7742. 1

work page internal anchor Pith review Pith/arXiv arXiv 2014
[24]

Mili ´cevi´c,General inverse theory for theU 4 norm, preprint (2026),https://arxiv.org/abs/26 01.01682

L. Mili ´cevi´c,General inverse theory for theU 4 norm, preprint (2026),https://arxiv.org/abs/26 01.01682. 1

work page 2026
[25]

Mili ´cevi´c,Quantitative inverse theorem for Gowers uniformity normsU 5 andU 6 inF n 2 to appear in Canadian Journal of Mathematics

L. Mili ´cevi´c,Quantitative inverse theorem for Gowers uniformity normsU 5 andU 6 inF n 2 to appear in Canadian Journal of Mathematics. 1

work page
[26]

Mili ´cevi´c,Quasipolynomial inverse theorem for theU 4(Fn p )norm, preprint (2024),https://arxiv

L. Mili ´cevi´c,Quasipolynomial inverse theorem for theU 4(Fn p )norm, preprint (2024),https://arxiv. org/abs/2410.08966

work page arXiv 2024
[27]

Rudin,Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No

W. Rudin,Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12. Inter- science Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. ix+285 pp. 14

work page 1962
[28]

Tao and V

T. Tao and V . Vu,Additive combinatorics, Cambridge Stud. Adv. Math., 105 Cambridge University Press, Cambridge, 2006. xviii+512 pp. 24

work page 2006
[29]

T. Tao, T. Ziegler,The inverse conjecture for the Gowers norm over finite fields in low characteristic, Ann. Comb.16(2012), 121–188. 1 INSTITUTO DECIENCIASMATEM ´ATICAS, CALLENICOL ´ASCABRERA13-15, MADRID28049, SPAIN Email address:pablo.candela@icmat.es UNIVERSIT ´EPARISCIT ´E, SORBONNEUNIVERSIT ´E, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE Email address:gonza...

work page 2012

[1] [1]

Bergelson, T

V . Bergelson, T. Tao and T. Ziegler,An inverse theorem for the uniformity seminorms associated with the action ofF ∞ p , Geom. Funct. Anal.19(2010), no. 6, 1539–1596. 1

work page 2010

[2] [2]

Antol ´ın Camarena, B

O. Antol ´ın Camarena, B. Szegedy,Nilspaces, nilmanifolds and their morphisms, preprint (2010),http: //arxiv.org/abs/1009.38252

work page arXiv 2010

[3] [3]

Candela,Notes on nilspaces: algebraic aspects, Discrete Anal., 2017, Paper No

P. Candela,Notes on nilspaces: algebraic aspects, Discrete Anal., 2017, Paper No. 15, 59 pp. 3, 4, 5, 6, 9, 10, 17

work page 2017

[4] [4]

Candela,Notes on compact nilspaces, Discrete Anal., 2017, Paper No

P. Candela,Notes on compact nilspaces, Discrete Anal., 2017, Paper No. 16, 57pp. 2, 4, 5, 6, 8, 9, 10, 11, 12

work page 2017

[5] [5]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyOn nilspace systems and their morphisms, Ergodic Theory Dynam. Systems40(2020), no. 11, 3015–3029. 6

work page 2020

[6] [6]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. Szegedy,On higher-order Fourier analysis in characteristicp, Ergodic Theory Dynam. Systems43(2023), no. 12, 3971-4040. 12

work page 2023

[7] [7]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyFree nilspaces, double coset nilspaces, and Gowers norms, preprint (2025),https://arxiv.org/abs/2305.11233. 17, 18, 22

work page arXiv 2025

[8] [8]

Candela, D

P. Candela, D. Gonz ´alez-S´anchez, B. SzegedyOn the inverse theorem in abelian groups of bounded torsion, preprint (2023),https://arxiv.org/abs/2311.13899. 15

work page arXiv 2023

[9] [9]

Candela, D

P. Candela, D. Gonz´alez-S´anchez, B. SzegedyAn inverse theorem for all finite abelian groups via nilmanifolds, preprint (2025),https://arxiv.org/abs/2512.17468. 1

work page arXiv 2025

[10] [10]

Candela, B

P. Candela, B. Szegedy,Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds, J. Reine Angew. Math.789(2022), 1–42. 2, 4, 6, 8, 10, 11, 13

work page 2022

[11] [11]

W. T. Gowers,A new proof of Szemer ´edi’s theorem, Geom. Funct. Anal.11(2001), no. 3, 465–588

work page 2001

[12] [12]

W. T. Gowers, L. Mili´cevi´c,A quantitative inverse theorem for theU 4 norm over finite fields, preprint (2017), https://arxiv.org/abs/1712.00241. 1

work page internal anchor Pith review Pith/arXiv arXiv 2017

[13] [13]

Green, T

B. Green, T. Tao,An inverse theorem for the GowersU 3-norm, Proc. Edinburgh Math. Soc. (1)51(2008), 73–153. 1

work page 2008

[14] [14]

Green, T

B. Green, T. Tao,The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math.175(2012), 465–540. 22

work page 2012

[15] [15]

Green, T

B. Green, T. Tao, T. Ziegler,An inverse theorem for the GowersU 4-norm, Glasgow Math. J.53(2011), 1–50. 1 26 PABLO CANDELA, DIEGO GONZ ´ALEZ-S ´ANCHEZ, AND BAL ´AZS SZEGEDY

work page 2011

[16] [16]

Green, T

B. Green, T. Tao, T. Ziegler,An inverse theorem for the GowersU s+1[N]-norm, Ann. of Math. (2)176 (2012), no. 2, 1231–1372. 1, 16, 18, 21, 22

work page 2012

[17] [17]

K. H. Hofmann, S. A. Morris,The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie al- gebras, pro-Lie groups, and connected locally compact groups. EMS Tracts Math., 2. European Mathematical Society (EMS), Z¨urich, 2007. 17

work page 2007

[18] [18]

B. Host, B. Kra,Nilpotent structures in ergodic theory, Math. Surveys Monogr., 236. American Mathematical Society, Providence, RI, 2018. 16, 19

work page 2018

[19] [19]

Jamneshan, O

A. Jamneshan, O. Shalom, T. Tao,Polynomial towers and inverse Gowers theory for bounded-exponent groups, preprint (2026),https://arxiv.org/abs/2601.00961. 1

work page arXiv 2026

[20] [20]

Jamneshan, T

A. Jamneshan, T. Tao,The inverse theorem for theU 3 Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches, Discrete Anal., 2023:11, 48 pp. 1, 4

work page 2023

[21] [21]

J. Leng, A. Sah, M. Sawhney,Quasipolynomial bounds on the inverse theorem for the GowersU s+1[N]- norm, prerpint (2024),https://arxiv.org/abs/2402.17994. 1

work page arXiv 2024

[22] [22]

Manners,Quantitative bounds in the inverse theorem for the GowersU s+1-norms over cyclic groups preprint (2018),http://www.arxiv.org/abs/1811.00718

F. Manners,Quantitative bounds in the inverse theorem for the GowersU s+1-norms over cyclic groups preprint (2018),http://www.arxiv.org/abs/1811.00718. 1

work page arXiv 2018

[23] [23]

Periodic nilsequences and inverse theorems on cyclic groups

F. Manners,Periodic nilsequences and inverse theorems on cyclic groupspreprint (2014),http://www. arxiv.org/abs/1404.7742. 1

work page internal anchor Pith review Pith/arXiv arXiv 2014

[24] [24]

Mili ´cevi´c,General inverse theory for theU 4 norm, preprint (2026),https://arxiv.org/abs/26 01.01682

L. Mili ´cevi´c,General inverse theory for theU 4 norm, preprint (2026),https://arxiv.org/abs/26 01.01682. 1

work page 2026

[25] [25]

Mili ´cevi´c,Quantitative inverse theorem for Gowers uniformity normsU 5 andU 6 inF n 2 to appear in Canadian Journal of Mathematics

L. Mili ´cevi´c,Quantitative inverse theorem for Gowers uniformity normsU 5 andU 6 inF n 2 to appear in Canadian Journal of Mathematics. 1

work page

[26] [26]

Mili ´cevi´c,Quasipolynomial inverse theorem for theU 4(Fn p )norm, preprint (2024),https://arxiv

L. Mili ´cevi´c,Quasipolynomial inverse theorem for theU 4(Fn p )norm, preprint (2024),https://arxiv. org/abs/2410.08966

work page arXiv 2024

[27] [27]

Rudin,Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No

W. Rudin,Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12. Inter- science Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. ix+285 pp. 14

work page 1962

[28] [28]

Tao and V

T. Tao and V . Vu,Additive combinatorics, Cambridge Stud. Adv. Math., 105 Cambridge University Press, Cambridge, 2006. xviii+512 pp. 24

work page 2006

[29] [29]

T. Tao, T. Ziegler,The inverse conjecture for the Gowers norm over finite fields in low characteristic, Ann. Comb.16(2012), 121–188. 1 INSTITUTO DECIENCIASMATEM ´ATICAS, CALLENICOL ´ASCABRERA13-15, MADRID28049, SPAIN Email address:pablo.candela@icmat.es UNIVERSIT ´EPARISCIT ´E, SORBONNEUNIVERSIT ´E, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE Email address:gonza...

work page 2012