{"id":"416bf70b-2089-4a10-aed4-478ab7b8eaf4","arxiv_id":"2505.15549","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Prime-weighted polynomial multiple ergodic averages converge pointwise almost everywhere for arbitrary k and distinct-degree integer polynomials, with r-variational estimates for every r > 2.","lead":"Mathematicians prove that averages of products of several outputs along prime-indexed polynomial times converge almost everywhere, for any number of polynomials with different degrees. This is the first such pointwise theorem for many polynomials, with strong quantitative variation bounds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reduction from Λ to the Cramér approximant Λ_N relies on the C0=100 estimate (4.10), asserted only in footnote 7; the missing derivation is load-bearing, though a short argument using Lemma 2.2(ii) and the C0=10 case of [35] may supply it.","rationale":"I read the paper as a serious and substantial advance: it gives the first pointwise almost everywhere result for general polynomial multiple ergodic averages along the primes, with a variational inequality, and the overall architecture is coherent. The reader's conditional verdict is reasonable, but the specific weakest assumption they flagged, the C0 = 100 case of (4.10), is not clearly fatal: it is very likely derivable from the C0 = 10 case of [35] together with the already-quoted Lemma 2.2(ii). I therefore partially agree with the reader: the concern is real as an omitted verification and as a load-bearing step, but it may be an expositional gap rather than a mathematical error. The point of maximum leverage remains (4.10), because (4.6) and then the entire reduction to the Cramér model depend on it; if that estimate failed, the theorem would require a different treatment of the exceptional endpoint. Since the missing derivation is short and checkable, and since no internal inconsistency or unsupported central construction was found, the appropriate verdict is unchanged: conditional acceptance pending verification of the numeric input and the normalization in the application of Lemma 2.3.","tokens_in":50905,"tokens_out":20188,"duration_ms":186454,"concrete_test":"Derive (4.10) for C0 = 100 explicitly: with z_C = exp(Log^{1/C}N), write ∥Λ − Λ_{z_100}∥_{u_{d_k+1}[N]} ≤ ∥Λ − Λ_{z_10}∥_{u_{d_k+1}[N]} + ∥Λ_{z_10} − Λ_{z_100}∥_{u_{d_k+1}[N]}, bound the first term using [35, Theorem 1.1] for C0 = 10 and the second using Lemma 2.2(ii); verify the total is O_M(⟨Log N⟩^{-M}) for every M. Then redo the proof of (4.11) with the normalized weight θ₀ = (Λ − Λ_N)/(C Log N), using Lemma 2.3 on θ₀, and check that the factor Log N is absorbed by taking M sufficiently large. If both calculations close, (4.6) is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proof reduces the von Mangoldt weight to the Cramér weight in Section 4.1 via (4.6), and the key estimate is (4.10): ∥Λ − Λ_N∥_{u_{d_k+1}[N]} ≲_{M,C0} ⟨Log N⟩^{-M} with Λ_N = Λ_{Cramér, exp(Log^{1/C0}N)} and C0 = 100. This estimate is used immediately to obtain (4.11), and without it the replacement of Λ by the structured Cramér model fails, leaving the theory without the needed minor/major arc reduction. The paper does not prove (4.10) for C0 = 100; footnote 7 only asserts that the methods of [35], which treats C0 = 10, extend to arbitrarily large C0. The assertion is plausible and probably correct: combining [35] with Lemma 2.2(ii) and the triangle inequality should yield (4.10), since exp(Log^{1/100}N) lies within the range allowed in Lemma 2.2(ii) and exp(−c Log^{1/100}N) decays faster than any power of Log N. A second, related omission is that (4.11) applies Lemma 2.3 to the weight Λ − Λ_N, which is not 1-bounded; one must first normalize by O(Log N) and then absorb the resulting logarithmic factor by choosing M large. Both steps are checkable but not written out, and they are exactly the hinge between the genuinely prime-weighted problem and the Cramér model.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a pointwise almost everywhere convergence theorem for von Mangoldt-weighted polynomial multiple ergodic averages along the primes. For any k≥1, any invertible measure-preserving transformation, any integer-coefficient polynomials of distinct degrees, and any bounded functions, the averages converge pointwise a.e.; moreover, a sharp r-variational estimate with r>2 is established. The proof builds a multilinear circle method: it reduces the von Mangoldt weight to a Cramér approximant via a little Gowers uniformity estimate, proves a Cramér-weighted inverse theorem and Weyl-type minor-arc inequality, and then splits the major-arc analysis into high-frequency, low-frequency small-scale, and low-frequency large-scale cases. The latter two cases are handled through a new multilinear Rademacher-Menshov inequality and an arithmetic multilinear estimate on the adelic integers. The paper also states auxiliary theorems for σ-finite systems and gives a reduction to the integer shift system via Calderón transference.","tokens_in":51266,"tokens_out":4860,"duration_ms":45584,"significance":"If the proof is correct, the result is a substantial advance: it settles the prime-weighted analogue of Bergelson's question for polynomials of distinct degrees in the single-transformation case, refines the norm-convergence theorem of Wooley–Ziegler, and extends the bilinear prime-weighted pointwise result of Krause–Mousavi–Tao–Teräväinen to arbitrary k with L∞ inputs. The paper introduces a multilinear circle method for von Mangoldt weights, including a Cramér-weighted inverse theorem, a multilinear Rademacher-Menshov inequality, and arithmetic multilinear estimates, and it is notable for working with inverse theorems having subpolynomial bounds. The proof structure is coherent, and the author is transparent about the main external inputs, which include the Ionescu–Wainger theorem [30], Teräväinen's generalized von Neumann theorem [53], and the uniformity estimate [35]. The reliance on recent preprints is a real caveat, but those results are used as tools rather than as hidden assumptions of the conclusion.","major_comments":[{"comment":"The reduction from the von Mangoldt function Λ to the Cramér approximant Λ_N is the hinge of the proof, and the required uniformity estimate (4.10) is imported from [35] with the parameter C0=100, although [35, Theorem 1.1] is stated for C0=10. Footnote 7 asserts that the methods of [35] extend to arbitrarily large C0, but no derivation is supplied in the manuscript. Since (4.11) and the rest of Section 4.1 depend on (4.10) with this specific choice, the paper should either prove (4.10) for general C0 or state it as a lemma with a proof. This is not a cosmetic omission: without the C0=100 uniformity, the replacement of Λ by Λ_N fails and the minor/major arc reduction is not justified.","section":"§4.1, Eq. (4.10), footnote 7"},{"comment":"In proving (4.11), Lemma 2.3 is applied to the weight θ = Λ − Λ_N, but Lemma 2.3 requires |θ| ≤ 1, whereas (2.18) only gives |Λ − Λ_N| ≲ ⟨Log N⟩. One can normalize by ⟨Log N⟩ and then absorb the affine factor by choosing M sufficiently large, but this step is not written out. As the text stands, the displayed bound does not follow directly from the cited lemma. Please add the normalization argument and make explicit how the parameter M is used after the normalization.","section":"§4.1, Eq. (4.11) and Lemma 2.3"},{"comment":"The proof of Theorem 3.2 is carried out in detail for j=1 and j=2, but the cases j=3,…,k are dispatched with the sentence \"Similarly, repeating the above process.\" Since Theorem 3.2 feeds directly into the weighted inverse theorem, the dual-function structure, and ultimately the minor-arcs estimate (3.4), the induction over j should be stated more precisely, in particular how the distinct-degree condition is used at each step and how the polynomial-coefficient bounds are preserved under the congruence-based factorization. This is currently a gap in exposition, though it appears repairable.","section":"§3.1.1, proof of Theorem 3.2 for j≥3"}],"minor_comments":[{"comment":"The word \"yeilds\" should be \"yields\" in the final sentence of the proof.","section":"§3, proof of Proposition 3.4"},{"comment":"The word \"non-nagetive\" should be \"nonnegative\" near the definition of the function φ.","section":"§6.3, proof of (6.18)"},{"comment":"The symbol C0 is used both for the fixed exponent in the Cramér approximant (3.3) and for the sufficiently large threshold in (4.5); these two uses should be distinguished to avoid confusion.","section":"§4.1, notation"},{"comment":"The notation switches between ℓ^q(Z) and L^q(Z) in the same statements, particularly in (4.2) and the surrounding text; the counting-measure convention should be stated explicitly and used consistently.","section":"Theorem 4.1 and its proof"},{"comment":"The deduction of Theorem 1.2 from Theorem 1.3 is delegated to [31, Proposition 3.2] and [29, Theorem 1.6]; a brief sentence explaining the required modifications for the Λ_N weight would improve self-containedness, since the weighted averages are not literally operators treated in those references.","section":"§1.4 and §4.1, derivations"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and, after revision, would be a strong contribution. The main concern is the missing derivation of the C0=100 uniformity estimate (4.10) and the unnormalized application of Lemma 2.3 in (4.11); both appear to be fixable locally. I recommend major revision rather than rejection because the central line of the argument is defensible and the load-bearing gaps are concrete and likely repairable within the manuscript's framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a genuine advance. It proves pointwise a.e. convergence and sharp r-variational estimates (r>2) for von Mangoldt-weighted polynomial multiple ergodic averages for any number k of polynomials with distinct degrees under a single invertible transformation. Before this, pointwise a.e. convergence was known only in the bilinear case (n,P(n)) (Krause-Mousavi-Tao-Teravainen) and norm convergence in general (Wooley-Ziegler). The paper closes that gap and gives the first k>=3 result.\n\nThe new machinery is real: a multilinear circle method for Cramer-weighted averages, a weighted inverse theorem with subpolynomial bounds, a multilinear Rademacher-Menshov inequality, and an arithmetic multilinear estimate. The structure is coherent and most lemmas are proved in detail. The authors are honest about what comes from previous work: the generalized von Neumann theorem [53], the Ionescu-Wainger theorem [30], the uniformity estimate (4.10) from [35], and the framework from [31]. These are external inputs with independent standing, not hidden assumptions.\n\nThe soft spots are real but, I think, repairable. The reduction from Lambda to the Cramer model depends on (4.10), the little Gowers estimate for C0=100, and footnote 7 only asserts that the methods of [35] extend from C0=10 to arbitrarily large C0. No derivation is given. That is the hinge between the actual prime-weighted problem and the structured model. The stress-test note is right that a proof can likely be assembled from Lemma 2.2(ii) and the C0=10 case, but the paper should write it out. Similarly, (4.11) applies Lemma 2.3 to Lambda - Lambda_N, which is not 1-bounded; normalizing by O(Log N) and absorbing the log by choosing M large is standard but needs to be visible. Both are checkable fixes rather than signs of a false theorem.\n\nOne editorial problem: the arXiv metadata lists three authors, while the title page lists only Renhui Wan. That needs sorting out before publication.\n\nMy overall read: the central claim is likely correct, the proof is long and rests on several recent preprints, and the load-bearing C0=100 estimate should be either proved or cited to a source that proves it. This paper deserves a serious referee despite the caveats. I would send it to review, and I would cite it in related work.","headline":"First pointwise a.e. theorem for prime-weighted polynomial multiple ergodic averages with k>=3; proof likely correct, but the C0=100 uniformity assertion needs to be written out.","tokens_in":51795,"tokens_out":2897,"would_cite":true,"duration_ms":26896,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A46","42A45","42A85","43A25","11B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves pointwise almost-everywhere convergence for von Mangoldt-weighted polynomial multiple ergodic averages of any order k, for polynomials of distinct integer degrees, with r-variational estimates for every r>2.","keywords":["polynomial multiple ergodic averages","pointwise convergence","von Mangoldt weight","primes","variational estimates","multilinear circle method","inverse theorem","little Gowers norms"],"falsifier":"Run the correlation computation behind estimate (4.10) with the parameter $C_0=100$: compute the little Gowers norm $\\|\\La-\\La_N\\|_{u_{d+1}[N]}$, a measure of correlation with polynomial phases, for a fixed degree $d$, at scales $N=10^m$ up to $10^{12}$, with $\\La_N$ the Cramér approximant at $\\exp(\\log^{1/100}N)$. The claim requires this to decay faster than any power of $\\log N$; any scale at which the decay is merely polynomial, or absent, would falsify the estimate on which the reduction from von Mangoldt to Cramér averages rests.","tokens_in":50709,"feed_emoji":"🧪","tokens_out":12737,"duration_ms":104615,"temperature":0.7,"pith_summary":"The paper establishes pointwise almost-everywhere convergence for averages of the form $$\\frac{1}{N}\\sum_{n=1}^{N}\\La(n) f_1($T^{{P_1(n)}}$x)\\cdots f_k($T^{{P_k(n)}}$x),$$ where $\\La$ is the von Mangoldt function, equal to $\\log p$ on prime powers and zero otherwise, $T$ is an invertible measure-preserving transformation of a probability space, and $P_1,\\ldots,P_k$ are integer polynomials of distinct degrees. This settles the prime-weighted analogue of a long-standing multiple-recurrence question for any number $k$ of polynomial iterates, and it does so quantitatively: the averages satisfy $r$-variational bounds for every $r>2$, not merely convergence. The result refines the earlier norm-convergence theorem for these averages to pointwise convergence and removes the restrictions of the previously known bilinear case, at the price of requiring the functions to lie in $L^\\infty(X)$. The paper's contribution is a multilinear circle method adapted to prime weights, whose distinguishing feature is that it works with inverse theorems that supply only subpolynomial bounds.","feed_headline":"Prime-weighted polynomial averages converge pointwise","feed_subtitle":"New multilinear circle method proves pointwise convergence for any number of distinct-degree polynomial iterates.","key_machinery":"The central object is the Cramér approximant $\\La_N(n)$, a weighted indicator of integers coprime to the product of all primes up to $\\exp(\\mathrm{Log}^{1/100}N)$, which stands in for the von Mangoldt function inside the averages. The argument is carried by four components: (i) a weighted multilinear inverse theorem and a multilinear Weyl inequality for Cramér-weighted averages, giving minor-arcs decay $2^{-cl}+\\exp(-c\\mathrm{Log}^{1/C_0}N)$ that is subpolynomial rather than polynomial; (ii) the multiplier theorem for canonical fractions, which compares discrete and continuous Fourier multipliers; (iii) a multilinear Rademacher–Menshov inequality, derived inductively from the bilinear case, used for the small-scale low-frequency case; and (iv) an arithmetic multilinear estimate on the adelic integers together with $p$-adic $L^2\\to L^{2s}$ improving estimates, used for the large-scale low-frequency case. The subpolynomial decay makes the earlier metric-entropy arguments inapplicable, and the new low-frequency estimates replace them.","core_discovery":"The paper's central claim, Theorem 1.1, is that for every $k\\ge 1$, every invertible measure-preserving transformation $T$ of a probability space $(X,\\nu)$, every family of integer-coefficient polynomials $P_1,\\ldots,P_k$ of distinct degrees, and every $f_1,\\ldots,f_k\\in L^\\infty(X)$, the weighted multiple ergodic averages $A^P_{N,\\La;X}(f_1,\\ldots,f_k)$ converge pointwise $\\nu$-almost everywhere as $N\\to\\infty$, and satisfy the $r$-variational estimate (1.5) for every $r>2$ and $0<q<\\infty$, with constants depending only on the polynomials, the lacunarity parameter, $r$, $q$, and $k$. Because the von Mangoldt function is supported on prime powers, this is equivalent to almost-everywhere convergence of the prime-weighted averages taken over primes $p\\le N$. The proof transfers the problem to the integer shift system and develops a multilinear circle method: a minor-arcs estimate for Cramér-weighted averages, a major-arcs estimate split into high-frequency, small-scale low-frequency, and large-scale low-frequency cases, and supporting harmonic-analysis and arithmetic estimates on the adelic integers.","pith_inferences":["Estimate (4.10), the reduction from the von Mangoldt weight to the Cramér approximant, is imported from the cited higher-uniformity work with $C_0=100$ by a footnote that asserts an extension beyond the stated $C_0=10$; no derivation of that extension appears here, so the full theorem as stated depends on an unverified uniformity claim.","The subpolynomial decay of the minor-arcs estimate explains the paper's restriction to a single transformation: the polynomial-decay arguments that handle commuting transformations in the unweighted setting do not survive once the prime weight is present, so an extension to commuting transformations would require a genuinely new low-frequency argument.","The requirement $f_1,\\ldots,f_k\\in L^\\infty$ enters through the norm-interchanging trick and the $p$-adic estimates in the large-scale low-frequency case; extending the theorem to $L^p$ inputs would require a different treatment of that case.","The arithmetic multilinear estimate on the adelic integers is a general statement about polynomial configurations on profinite groups with one function missing low frequencies, so it may be reusable for other prime-weighted or multiplicative-weight problems in ergodic theory and additive combinatorics."],"forward_implications":["The prime-weighted averages (1.4) converge pointwise almost everywhere for any number $k$ of polynomial iterates with distinct integer degrees, removing the bilinear restriction of the earlier pointwise result.","The $r$-variational estimates for every $r>2$ provide quantitative control over the oscillations of the averages along lacunary sequences, not just eventual convergence.","Since pointwise convergence on a probability space implies norm convergence by dominated convergence, the earlier norm-convergence theorem for these averages is recovered as a corollary.","With the necessary adjustments, the same method also gives pointwise almost-everywhere convergence of the unweighted distinct-degree polynomial multiple ergodic averages.","The fact that the circle method can be run with inverse theorems that supply only subpolynomial bounds is a structural gain: it suggests that other multilinear problems without polynomial-bounded inverse theorems are now approachable."],"supporting_citations":[{"why":"Supplies the critical little Gowers-norm approximation of the von Mangoldt function by the Cramér approximant, estimate (4.10); the paper uses it with $C_0=100$ and cites a footnote asserting the extension.","marker":"[35]"},{"why":"Establishes the bilinear prime-weighted pointwise result and supplies the Cramér and Heath-Brown approximant estimates and the regularity bounds used in the reduction.","marker":"[32]"},{"why":"Provides the multilinear circle method for unweighted averages, the multiplier theorem for canonical fractions, and the continuous multilinear Weyl inequality used for the high-frequency major-arcs case.","marker":"[30]"},{"why":"Supplies the bilinear circle method, the bilinear Rademacher–Menshov inequality, and the adelic sampling and Shannon framework that the new multilinear estimates generalize.","marker":"[31]"},{"why":"The inverse theorem for polynomial progressions is the input that yields the unweighted inverse theorem behind the weighted inverse theorem.","marker":"[47]"},{"why":"The generalized von Neumann theorem lets the proof pass from von Mangoldt-weighted averages to Cramér-weighted averages once a Gowers-uniformity estimate is available.","marker":"[53]"},{"why":"Established norm convergence of these prime-weighted polynomial ergodic averages; the pointwise theorem refines it.","marker":"[57]"},{"why":"The transference principle used to pass from the integer shift system to arbitrary measure-preserving systems.","marker":"[11]"}],"fun_headline_variants":["Pointwise convergence for prime-weighted ergodic averages","Multilinear circle method proves pointwise a.e. convergence","Distinct-degree polynomial averages: pointwise a.e. result","Primes in ergodic averages: almost-everywhere convergence","New multilinear tool proves pointwise convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem stands on the assumption that the Cramér approximant, a simple weight built from the small primes, can be substituted for the von Mangoldt function at the very slow scale $\\exp((\\log N)^{1/100})$ with an error that decays faster than every power of $\\log N$ in the little Gowers norm, which measures correlation with polynomial phases; the paper takes the needed uniformity with $C_0=100$ from a cited result, asserting in a footnote that the methods extend, without giving the derivation.","fun_headline_variants_meta":{"raw":{"variants":["Pointwise convergence for prime-weighted ergodic averages","Multilinear circle method proves pointwise a.e. convergence","Distinct-degree polynomial averages: pointwise a.e. result","Primes in ergodic averages: almost-everywhere convergence","New multilinear tool proves pointwise convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1925,"prompt_tokens":1105,"completion_tokens":820,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":721,"completion_tokens_details":{"reasoning_tokens":742}},"tokens_in":721,"tokens_out":820,"duration_ms":7106,"temperature":1.0,"reasoning_tokens":742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:14:54.255996+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the correlation computation behind estimate (4.10) with the parameter $C_0=100$: compute the little Gowers norm $\\|\\La-\\La_N\\|_{u_{d+1}[N]}$, a measure of correlation with polynomial phases, for a fixed degree $d$, at scales $N=10^m$ up to $10^{12}$, with $\\La_N$ the Cramér approximant at $\\exp(\\log^{1/100}N)$. The claim requires this to decay faster than any power of $\\log N$; any scale at which the decay is merely polynomial, or absent, would falsify the estimate on which the reduction from von Mangoldt to Cramér averages rests.","supporting_citations":[{"cited_title":"Matom¨ aki, X","cited_arxiv_id":null,"evidence_quote":"Supplies the critical little Gowers-norm approximation of the von Mangoldt function by the Cramér approximant, estimate (4.10); the paper uses it with $C_0=100$ and cites a footnote asserting the extension."},{"cited_title":"Krause, H","cited_arxiv_id":null,"evidence_quote":"Establishes the bilinear prime-weighted pointwise result and supplies the Cramér and Heath-Brown approximant estimates and the regularity bounds used in the reduction."},{"cited_title":"Krause, M","cited_arxiv_id":null,"evidence_quote":"Supplies the bilinear circle method, the bilinear Rademacher–Menshov inequality, and the adelic sampling and Shannon framework that the new multilinear estimates generalize."},{"cited_title":"Peluse, Bounds for sets with no polynomial progressions, Forum Math","cited_arxiv_id":null,"evidence_quote":"The inverse theorem for polynomial progressions is the input that yields the unweighted inverse theorem behind the weighted inverse theorem."},{"cited_title":"Wooley, T","cited_arxiv_id":null,"evidence_quote":"Established norm convergence of these prime-weighted polynomial ergodic averages; the pointwise theorem refines it."},{"cited_title":"Calder´ on, Ergodic theory and translation invariant operators, Proc","cited_arxiv_id":null,"evidence_quote":"The transference principle used to pass from the integer shift system to arbitrary measure-preserving systems."}],"review_version":1}