arXiv:2607.29368v1

Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials

Andreas Koutsogiannis, Borys Kuca, Wenbo Sun

math.DSmath.CO37A3011B3028D0537A44

Paper metadata

Original submission: July 31, 2026
Claimed MSC: 37A30, 11B30, 28D05, 37A44
arXiv categories: math.DS, math.CO
Paper license: CC BY 4.0 ↗

Abstract

We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving $\mathbb{Z}^D$ -systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if $T_1,\ldots,T_\ell$ are totally ergodic and generate a 2-step nilpotent group, then $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_1^n f_1 \cdots T_\ell^{n^\ell}f_\ell = \prod_{j=1}^{\ell}\int f_j\,dμ$ in the $L^{2}$ norm for all bounded functions $f_{1},\dots,f_{\ell}$ ; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szemeédi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and $\mathbb Z^D$ -systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

Theorems 1.5, 1.6, 1.13–1.16, 1.18, and 1.20 were checked. They respectively establish joint ergodicity for totally ergodic distinct-degree iterates, the required characteristic-factor bound, the commuting-system criterion, the two counterexamples and positive two-step criterion, the multidirectional seminorm estimate, and nilsystem equidistribution. The hypotheses in the individual statements are preserved throughout their deductions.

Theorems 1.5 and 1.6Correct

Joint ergodicity and characteristic-factor control for distinct-degree iterates

Pages 4–6 · Theorems 1.5–1.6 · arXiv:2607.29368v1

Theorem 1.6 gives a Host–Kra seminorm of bounded order that controls every distinct-degree polynomial average generated by a two-step nilpotent action. When the individual transformations are totally ergodic, this control reduces the average to the product of the integrals, exactly the joint-ergodicity conclusion of Theorem 1.5. The degree ordering, two-step nilpotence, boundedness of the functions, and total-ergodicity hypotheses are retained at the points where each is required.

Theorems 1.13–1.16Correct

Joint-ergodicity criteria and the two affine-nilsystem counterexamples

Pages 7–9 and 55–57 · Theorems 1.13–1.16 · arXiv:2607.29368v1

For commuting integer actions, Theorem 1.13 combines product-action ergodicity with ergodicity of every difference family, and the proof derives the converse from the uniform seminorm control. Theorem 1.15 correctly simplifies the criterion for two-step nilpotent actions with pairwise distinct polynomial degrees. The explicit affine transformations in Theorems 1.14 and 1.16 have the printed commutator structure; their toral coordinate calculations establish the claimed product and pairwise ergodicity while a surviving vertical character witnesses failure of joint ergodicity.

Theorems 1.18 and 1.20Correct

Multidirectional seminorm control and nilsystem equidistribution

Pages 9–10 and 50–55 · Theorems 1.18 and 1.20 · arXiv:2607.29368v1

Theorem 1.18 uses all coefficient-difference subgroups H(j,j') in a symmetric box seminorm and gives the vanishing implication stated in the introduction. For Theorem 1.20, total ergodicity forces the nilmanifold to be connected; Leibman's criterion then reduces equidistribution to the horizontal torus, where independence of the polynomials and total ergodicity eliminate every nontrivial character. This proves almost-everywhere equidistribution in the full product nilmanifold with the stated quantifiers.

02Proofs4 reported findingsCorrect

Seminorm smoothing, nilpotent PET induction, and Host–Kra factor comparison. The proof first derives the multidirectional box-seminorm estimate of Theorem 1.18 and transfers it to the single-transformation seminorms needed in Theorem 1.6. The two-step nilpotent PET induction decreases the declared weight vector under every van der Corput operation, so it terminates at the base systems. Section 8 then compares the relevant Host–Kra factors under conjugate nilpotent generators. With the two mechanical subscript corrections recorded below, the factor maps intertwine the announced actions and the seminorm implication closes without an omitted degree case.

Sections 5–8Correct and complete

Seminorm smoothing, nilpotent PET induction, and Host–Kra factor comparison

Pages 24–50 · Sections 5–8 · arXiv:2607.29368v1

The proof first derives the multidirectional box-seminorm estimate of Theorem 1.18 and transfers it to the single-transformation seminorms needed in Theorem 1.6. The two-step nilpotent PET induction decreases the declared weight vector under every van der Corput operation, so it terminates at the base systems. Section 8 then compares the relevant Host–Kra factors under conjugate nilpotent generators. With the two mechanical subscript corrections recorded below, the factor maps intertwine the announced actions and the seminorm implication closes without an omitted degree case.

Sections 9–10Correct and complete

Equidistribution and final joint-ergodicity deductions

Pages 50–57 · Sections 9–10 · arXiv:2607.29368v1

The horizontal-character reduction in Theorem 1.20 treats connected components before applying polynomial equidistribution, and Proposition 9.3 supplies the obstruction needed in the disconnected case. The final section combines this nilsystem calculation with the characteristic-factor inclusion: functions orthogonal to the relevant factors contribute zero, while the nilsystem components have the required product limit. The same criterion, used in both directions, yields Theorems 1.5, 1.13, and 1.15.

Proposition 8.2Typo · no status impact

The factor-map index is off by one

Page 47 · proof of Proposition 8.2 · arXiv:2607.29368v1

The systems in the preceding sentence are conjugated by the transformation indexed $\ell+1$ , but the displayed factor map is printed with the transformation indexed $\ell$ . Replacing that subscript by $\ell+1$ is forced by the displayed conjugacy and leaves the proof unchanged.

Theorem 1.5Typo · no status impact

The Host–Kra factor subscript is mismatched

Page 56 · proof of Theorem 1.5 · arXiv:2607.29368v1

Immediately after establishing an inclusion into the factor indexed $s'$ , the text calls the system with the factor indexed $s$ an $s'$ -step system. The preceding inclusion and the next invocation require the factor indexed $s'$ ; this unique subscript correction is harmless.

03Novelty0 reported findingsNo non-novelty findings

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Paper: arXiv:2607.29368v1
Authors listed: Andreas Koutsogiannis, Borys Kuca, Wenbo Sun
Audit date: August 18, 2026

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Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Joint ergodicity and characteristic-factor control for distinct-degree iterates

Joint-ergodicity criteria and the two affine-nilsystem counterexamples

Multidirectional seminorm control and nilsystem equidistribution

Seminorm smoothing, nilpotent PET induction, and Host–Kra factor comparison

Equidistribution and final joint-ergodicity deductions

The factor-map index is off by one

The Host–Kra factor subscript is mismatched

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