{"id":"1a8bc1d1-c1f2-416e-b1b4-c876e75ad465","arxiv_id":"2601.03999","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Polynomial-phase ergodic and singular-integral averages on homogeneous Lie groups satisfy uniform r-variation bounds for every r>2.","lead":"This paper proves quantitative polynomial Wiener–Wintner theorems: ergodic averages and singular-integral averages over homogeneous nilpotent Lie groups converge with uniform control in polynomial phase. It extends a classical theorem of ergodic theory to a far more general group-theoretic and analytic setting using a companion polynomial Carleson theorem.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorem rests on unproved companion theorem [4] and on Lemma 5.4's van der Corput verification; no formal certificate is provided.","rationale":"The reader's verdict was CONDITIONAL, and our stress-test identifies the same load-bearing concern: the main theorem is a corollary of an external companion theorem that is not proved here. We checked the internal reductions (Sections 2, 3, 4) and found no obvious gap: the linearization arguments, the kernel estimates using Lemmas 2.4 and 2.5, the approximation argument in Section 3, and the sparse domination in Section 4 appear sound. The verification of the compatibility and cancellation conditions for homogeneous Lie groups (Lemmas 5.2 and 5.4) is plausible and uses finite-dimensional polynomial norm equivalence and van der Corput estimates. We do not see an internal inconsistency. However, because Theorem 2.2 is the engine of the paper and is cited from a preprint by the same authors, the result is genuinely conditional. The claim of formal verification is not evidenced by a concrete artifact in this arXiv submission; it references only a blueprint. This does not make the mathematics false, but it justifies the CONDITIONAL verdict. We therefore recommend no change to the reader's verdict.","tokens_in":19073,"tokens_out":14200,"duration_ms":117110,"concrete_test":"Obtain the proof or the machine-checked statement of Theorem 1.1 from the companion paper [4] and verify that the compatible ε-cancellative family of §1.2.2–1.2.3 satisfies exactly the hypotheses of that theorem. If no formal certificate exists, independently re-derive Theorem 2.2 for the specific kernels constructed in §2.5 (equations (2.15)–(2.20)) and confirm the constants are uniform in the linearization parameters J, w, u_j. Additionally, expand the proof of Lemma 5.4 by giving the full statement of [35, Lemma A.1] and deriving the exponent ε=1/(d d_R) step-by-step for a sample homogeneous group such as the Heisenberg group.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3, and hence Corollary 1.4 and Theorems 1.5–1.6, are deduced from the generalized polynomial Carleson theorem (Theorem 2.2) quoted from the companion preprint [4]. This paper does not prove that theorem; Section 2 only checks that the kernels produced by the linearization procedure are one-sided β-kernels and that their nontangential maximal operators are L^2-bounded, then invokes [4] as a black box. If [4] has a gap, or if its hypotheses differ from the 'compatible ε-cancellative' conditions assumed here, nothing in this paper salvages the result. The verification of the ε-cancellative condition (1.5) for homogeneous Lie groups (Lemma 5.4) is a second load-bearing step: it relies on a van der Corput lemma from [35] and produces the specific exponent ε=1/(d d_R), but the paper does not give the details of [35, Lemma A.1] nor an independent proof of its applicability. The claim in §1.4.1 that [4] is 'formalized, i.e., verified by computer' is supported only by a blueprint of a formalization project [3], not by a certificate in this submission. Thus the central claim is internally coherent but conditionally dependent on an external unverified theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves quantitative polynomial Wiener-Wintner theorems in a very general setting: for d-dimensional metric measure spaces satisfying a restrictive volume measure condition, for compatible ε-cancellative families of modulations, and for homogeneous Lie groups with Leibman polynomial phases. The main estimate, Theorem 1.3, gives r-variation bounds (r>2) in the radial truncation parameter, uniformly over all admissible polynomial modulations, for both averages and singular integrals with cancellation kernels. The proof is a reduction: Theorem 2.2, quoted from the companion preprint [4], supplies the core generalized polynomial Carleson bound; [34] supplies the nontangential maximal function input; Sections 2–4 perform linearization, approximation of sharp cutoffs by smooth cutoffs, and sparse-domination extension to all p∈(1,∞). Section 5 verifies that homogeneous Lie groups and their Leibman polynomials satisfy the hypotheses, and Section 6 derives a corollary for quadratic polynomials into unitary groups.","tokens_in":19381,"tokens_out":7618,"duration_ms":73062,"significance":"If the companion theorem [4] is valid, the paper achieves a substantial unification: it recovers and extends previous polynomial Wiener–Wintner results to actions of homogeneous Lie groups, with uniform r-variation control and with singular-integral weights. The organization is explicit and the local arguments (linearization, cutoff approximation, sparse domination, transference) are coherent. The paper contains no fitted parameters and the reduction to the established machinery of [34] and [24] is natural. Its main weakness is external dependence: the load-bearing Carleson bound is not proved here, and the claim in §1.4.1 that [4] is computer-verified is not supported by the cited blueprint. The significance is therefore real but conditional on [4].","major_comments":[{"comment":"The central estimate is a direct consequence of Theorem 2.2, which is quoted from the companion preprint [4] and is not proved in this manuscript. The paper only checks that the linearized kernels are one-sided β-kernels and that the nontangential maximal operator is L²-bounded (Proposition 2.3). Thus the main theorem is conditional on an external, currently unpublished result by overlapping authors. This should be stated prominently in the introduction, and either a proof or a detailed and self-contained statement of Theorem 2.2 should be included before final acceptance.","section":"§1.2, Theorem 1.3 and §2.3, Theorem 2.2"},{"comment":"The verification of the ε-cancellative condition (1.5) for homogeneous Lie groups is load-bearing for Corollary 1.4, yet it invokes [35, Lemma A.1] without stating that lemma or its hypotheses. The reduction from the d_B oscillation to the parameter η=(1+d_B(f,g))^{-1/d_R} and the use of the C^{0,1} norm need to be written out. The specific exponent ε=1/(d d_R) should be derived explicitly from the stated van der Corput estimate. As written, a referee cannot verify the applicability of [35, Lemma A.1] to this non-abelian, polynomially distorted setting.","section":"§5, Lemma 5.4"},{"comment":"The assertion that the theorem in [4] is “formalized, i.e., verified by computer” is not supported by [3]. Reference [3] is a blueprint for the formalization of Carleson’s theorem on convergence of Fourier series, not a proof certificate for the generalized Carleson theorem on doubling metric measure spaces used here. This overstates the available evidence for [4] and should be corrected or removed, as it currently serves as an unjustified confidence boost for the main external input.","section":"§1.4.1"}],"minor_comments":[{"comment":"Theorem 1.2 is stated for p∈(1,∞], but Theorem 1.5, from which it is said to follow, is stated for p∈(1,∞). The p=∞ case does follow from the p=2 estimate on a probability space, but the implication should be stated explicitly.","section":"Theorem 1.2 vs Theorem 1.5"},{"comment":"In Lemma 2.4, the condition in (2.22) says γ≤min{1/2,α}, but the proof uses γ≤1/2 and later γ≤α. This is consistent, but it would help to define γ once and state both constraints before the proof.","section":"§2.5, Lemma 2.4"},{"comment":"The notation ‘x−y’ denotes the abelian group law on R^n while the group law is ‘◦’; this is introduced in Lemma 5.3 but should be flagged again in Lemma 5.4 to avoid confusion with the group inverse.","section":"§5, Lemma 5.3 and Lemma 5.4"},{"comment":"The Banach space Y of functions on Q×(0,∞) is defined with the norm sup_Q (|G(Q,1)|+||G(Q,·)||_{V^r_u}); it would be helpful to state explicitly that the supremum over Q is taken before the variation norm, since this is the space used in the sparse-dominated operator T.","section":"§4.1"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue for me is the dependence on [4]. The present paper is a well-organized reduction, but the core Carleson estimate is neither proved nor independently verified in this submission. The §1.4.1 formalization claim should not be used as a substitute. I would be willing to accept after the companion preprint is available in a stable, verifiable form and the local verifications in Section 5 are expanded."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a solid, useful paper. It does what it says: quantitative polynomial Wiener–Wintner theorems with r-variation control for ergodic averages and singular integral weights, on homogeneous Lie groups. That genuinely extends Lacey–Terwilleger and Oberlin et al. from linear phases on R to Leibman polynomials on nilpotent Lie groups. The proof is a sequence of reductions—linearize the variation norm, invoke the generalized polynomial Carleson theorem from the companion paper [4], replace smooth cutoffs by indicators via a standard comparison, and extend the exponent range via Lorist’s sparse domination. Each step is written out, and the checks of the hypotheses, especially compatibility and ε-cancellation for the modulation family, are handled explicitly. The paper also states its own limits clearly: §1.4.2 says non-homogeneous nilpotent groups and lattices are not covered, which is honest.\n\nWhere I part ways slightly with the reader: the stress-test note is right that the central engine [4] is not proved here, and the claim that it is “verified by computer” is supported only by a blueprint, not a certificate in this submission. That is a soft spot, and the authors should either provide the formal artifact or soften the claim. But I would not call the dependency circular or a flaw in the argument. The paper is transparent about the dependency, and the companion theorem is a natural continuation of Thiele’s program. The second load-bearing step, Lemma 5.4 verifying cancellation via van der Corput, is a bit compressed—it leans on [35, Lemma A.1] without reproducing it—but the argument is plausible and standard. So the main theorems are conditionally correct. If [4] holds up, this paper is a clean extension; if [4] has a gap, nothing here rescues it. That is a real risk, but not a reason to desk-reject.\n\nI also agree with the reader that there are no fitted parameters or invented entities. The heavy same-author citation is appropriate because [4] is the actual engine; that is how the literature works.\n\nBottom line: this deserves a serious referee. Send it to peer review, ideally with a referee who knows [4] and can judge the dependency. The paper is honest, technically careful, and advances a meaningful line. I would probably cite it once the companion preprint is settled.","headline":"A clean, honest reduction of polynomial Wiener–Wintner theorems to a companion Carleson theorem; the result is real, but the load-bearing engine lives in an unreleased preprint.","tokens_in":19889,"tokens_out":2498,"would_cite":true,"duration_ms":24875,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A45","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Ergodic averages over polynomial phases get uniform variation bounds","keywords":["Wiener–Wintner theorem","polynomial phases","r-variation norms","homogeneous Lie groups","nilpotent groups","singular integrals","ergodic averages","Carleson theorem"],"falsifier":"Find a doubling metric measure space and a compatible, ε-cancellative modulation family for which the maximally modulated singular integral T has unbounded norm on L^q for some 1<q≤2, or exhibit a homogeneous Lie group with a degree-d Leibman polynomial family for which the cancellation estimate (1.5) fails at some scale; either would refute Theorem 1.3 and its corollaries.","tokens_in":18947,"feed_emoji":"📈","tokens_out":7911,"duration_ms":66087,"temperature":0.7,"pith_summary":"This paper proves that the classical Wiener–Wintner theorem—uniform pointwise convergence of ergodic averages over all frequencies—extends to a quantitative, polynomial form for measure-preserving actions of homogeneous Lie groups. The central result is an L^p estimate, for p∈(1,∞) and r>2, on the r-variation in the ball radius of modulated averages and truncated singular integrals, uniformly over a compatible family of polynomial phases. This yields polynomial Wiener–Wintner theorems with explicit rate control for both plain ergodic averages and averages with singular-integral weights, and for quadratic phases that take values in unitary groups. The proof is a reduction: the variational estimates are deduced from a generalized polynomial Carleson theorem on doubling metric measure spaces, together with a transference principle and a sparse-domination step.","feed_headline":"Ergodic averages over polynomial phases get uniform variation bounds","feed_subtitle":"For homogeneous Lie groups, rate of convergence is uniform in the polynomial for averages and singular integrals.","key_machinery":"The argument is carried by three pieces. First, the r-variation norm, r>2, quantifies uniformity of convergence in the truncation parameter R (or u); this is the quantitative content. Second, the modulation family must be compatible and ε-cancellative—conditions that make the family behave like a finite-dimensional, well-separated set of phases and directly enable the oscillatory-cancellation estimate (1.5). Third, the decisive input is the generalized polynomial Carleson theorem: restricted weak-type L^q bounds for maximally modulated singular integrals on doubling metric measure spaces, which the paper invokes from its companion work. The proof then removes smooth cutoffs by an approximati","core_discovery":"On the paper's own terms, the central claim is Theorem 1.3: for any d-dimensional metric measure space carrying a compatible, ε-cancellative collection of modulation functions Q, and for p∈(1,∞), r>2, α∈(0,1], the supremum over Q of the r-variation in the truncation parameter of the modulated averages A_R(Q,f) is bounded in L^p by C∥f∥_p, and the same holds for truncated singular integrals S_u(K,Q,f) with kernels satisfying a cancellation condition. Specializing Q to the collection of real-valued Leibman polynomials of degree at most d on a homogeneous Lie group (Corollary 1.4) and applying the transference principle, the authors obtain quantitative polynomial Wiener–Wintner theorems for mea","pith_inferences":["Given that the companion Carleson theorem is computer-verified, a natural next step is a fully formalized proof of the transference and sparse-domination steps, which would make the ergodic theorems themselves machine-checked.","The paper's structure suggests that any modulation family satisfying compatibility and ε-cancellation automatically yields quantitative Wiener–Wintner theorems; a testable extension is to construct such families in non-doubling or non-homogeneous settings to see where the doubling requirement is truly needed.","The unitary-group case for quadratic phases is a template: if the unknown structure of higher-degree universal polynomial groups is divisible and solvable, as the paper notes, the same argument would prove the analogue for all degrees.","Because the proof only needs the cancellation estimate at scales where a Euclidean-type coordinate bound holds, a large-scale Wiener–Wintner theorem on general nilpotent Lie groups is plausible, as the paper itself expects."],"forward_implications":["The classical Wiener–Wintner theorem is extended in three ways at once: the acting group is a homogeneous (hence nilpotent) Lie group, the phases are polynomial rather than linear, and the convergence is quantified by finiteness of an r-variation norm uniform in the phase.","Pointwise convergence of polynomial ergodic averages holds with a single null set valid for every polynomial phase of bounded degree on these groups, with a uniform L^p bound on the r-variation.","The same quantitative uniform statement holds for ergodic averages with a singular-integral weight, including convolution kernels on the group.","For quadratic phases into unitary groups, a Wiener–Wintner statement with bounded smooth test functions follows (Corollary 1.7)."],"fun_headline_variants":["Uniform variation bounds for polynomial ergodic averages and singular integrals","Polynomial Wiener-Wintner theorems with uniform variation bounds","Quantitative Wiener-Wintner results for nilpotent Lie group actions","Uniform Lp bounds for polynomial phase averages and singular integrals","Sharp variation control for polynomial ergodic averages on groups"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The main estimate inherits the generalized polynomial Carleson theorem from the companion paper—restricted weak-type L^q bounds for maximally modulated singular integrals on doubling metric measure spaces—which is assumed, not proved here, and the ε-cancellation condition for the modulation family must hold; if either fails, Theorem 1.3 does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Uniform variation bounds for polynomial ergodic averages and singular integrals","Polynomial Wiener-Wintner theorems with uniform variation bounds","Quantitative Wiener-Wintner results for nilpotent Lie group actions","Uniform Lp bounds for polynomial phase averages and singular integrals","Sharp variation control for polynomial ergodic averages on groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000486,"raw_usage":{"total_tokens":2159,"prompt_tokens":595,"completion_tokens":1564,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":1495}},"tokens_in":339,"tokens_out":1564,"duration_ms":11238,"temperature":1.0,"reasoning_tokens":1495,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:09:15.632128+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a doubling metric measure space and a compatible, ε-cancellative modulation family for which the maximally modulated singular integral T has unbounded norm on L^q for some 1<q≤2, or exhibit a homogeneous Lie group with a degree-d Leibman polynomial family for which the cancellation estimate (1.5) fails at some scale; either would refute Theorem 1.3 and its corollaries.","supporting_citations":[],"review_version":1}