{"id":"cb56714f-145f-4c9b-bf8c-1fa2a0d15f30","arxiv_id":"1908.04521","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The orthogonal projection onto Helson matrices, and every multiplicative weighted version of it, is unbounded on the Schatten class S_q for every 1 ≤ q ≠ 2 < ∞.","lead":"Helson matrices, whose entries depend only on the product of the row and column indices, are the multiplicative cousins of Hankel matrices. This paper proves that the natural averaging projection onto Helson matrices, which is perfectly behaved in the Hilbert-Schmidt class, becomes unbounded in every other Schatten class S_q for 1 ≤ q ≠ 2 < ∞.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's proof normalizes test matrices to unit Schatten norm, which rules out the multiplicative hypothesis used to justify the tensor-product factorization; the step is repairable but is a genuine gap.","rationale":"The reader's weakest_assumption concerned the external identity (15) and the implicit verification of the Section 2.3 convergence conditions. That verification does succeed for the constructed operators, so I do not share that concern. However, the same tensor-product step contains a different, more concrete gap: the proof requires the chosen factors A_p to form a multiplicative matrix, but the unit-normalization imposed on A_p is incompatible with the condition ⟨A_p e1, e1⟩ = 1 that multiplicativity demands. A unit-norm operator satisfying that condition is forced to be rank-one with singular value 1, in which case the local projection P_{φ_p} fixes it and the ratio ‖P_{φ_p}A_p‖/‖A_p‖ equals 1, not 1+δ. Therefore the factorization P_Φ A = ⊗_p P_{φ_p}A_p, as justified in the paper via the multiplicative assumption, does not apply to the constructed A. The factorization itself is true for arbitrary tensor products, and the proof can be fixed by removing the normalization and using the ratios, so the central claim is likely correct. Because a nontrivial repair is needed in the written proof, I would make acceptance conditional on supplying that justification or adjusting the normalization step.","tokens_in":8663,"tokens_out":45083,"duration_ms":403461,"concrete_test":"Set q = 1 and take the uniform weight φ(i,j) = 1/(k+1), so φ0 = φ1 = φ2 = 1/3. Check whether any unit-norm A_p with (A_p)_{0,0} = 1 can satisfy ‖P_φ A_p‖_S1 ≥ 1 + δ with the δ from Lemma 4. Such an A_p must be rank-one with singular value 1, hence P_φ A_p = A_p and its S1-norm is 1; this confirms the proof's normalization step breaks the multiplicative hypothesis. Then recompute the final bound for Theorem 2 without normalizing A_p, using the ratios from Lemma 4 directly; if the inequality ‖P_Φ A‖_Sq ≥ (1+δ)^N ‖A‖_Sq still follows, the gap is repairable.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 2 (Section 3), the authors first suppose A = ⊗_p A_p is a multiplicative matrix, which by Section 2.3 requires ⟨A_p e1, e1⟩ = 1 for every p. They then fix N and choose, for p ≤ p_N, A_p with ‖A_p‖_Sq = 1 and ‖P_{φ_p}A_p‖_Sq ≥ 1 + δ. These two requirements are incompatible: if ‖A_p‖_Sq = 1 and ⟨A_p e1, e1⟩ = 1, then e1 is a singular vector with singular value 1 and all other singular values vanish, so A_p is a rank-one projection (up to phase) equal to H_{e0}. Consequently P_{φ_p}A_p = A_p and ‖P_{φ_p}A_p‖_Sq = 1, contradicting the required ≥ 1 + δ. Hence the A_p obtained by normalizing the witnesses from Lemma 4 do not satisfy the multiplicative hypothesis, and the factorization P_Φ A = ⊗_p P_{φ_p}A_p is not justified by the stated 'multiplicative matrix' route. The factorization is in fact valid for arbitrary tensor products because the divisor sum factors over primes, and the proof can be repaired by not normalizing A_p and using the ratios directly; but as written, the key estimate rests on an unproved step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the orthogonal projection from the Hilbert-Schmidt class S_2 onto the subspace of Hilbert-Schmidt Helson matrices, and more generally the weighted averaging projections P_Phi defined in (9) with a non-negative multiplicative weight Phi satisfying the normalization condition (8). The main theorem states that for every 1 <= q != 2 < infinity, every such P_Phi is unbounded on the Schatten class S_q; in particular, the uniform averaging projection (5) is unbounded on S_q for all q != 2. The proof introduces an explicit finite-dimensional lower bound (Lemma 4) showing that any Hankel-type weighted averaging projection has S_q norm at least 1+delta, and then transfers this bound to infinitely many tensor factors via an infinite tensor product representation of ell^2(N). Additional results assert the nonexistence of bounded projections onto Helson matrices in the spaces of compact and bounded operators, and a duality failure for Helson matrices in Schatten classes.","tokens_in":8764,"tokens_out":15190,"duration_ms":138587,"significance":"If the results are correct, the paper settles a natural multiplicative analogue of Peller's theorem on Hankel projections: unlike the Hankel case, the natural projection onto Helson matrices is unbounded for every q != 2, and no non-negative multiplicative reweighting of the averaging projection can repair this. The finite-dimensional computations in Lemma 4 are explicit and self-contained, and the uniform lower bound over all admissible weights is a notable strength. The tensor-product reduction is elegant and connects the problem to known results on multiplicative matrices. The paper is concise and the main ideas are transparent, though one step in the proof of Theorem 2 requires repair, as detailed below.","major_comments":[{"comment":"The proof chooses, for each prime p <= p_N, an operator A_p with ||A_p||_{S_q}=1 and ||P_{phi_p} A_p||_{S_q} >= 1+delta, and then states that A = tensor_p A_p is a multiplicative matrix. This is inconsistent with the definition in Section 2.3: if ||A_p||_{S_q}=1 and <A_p e_1, e_1>=1 (the condition for A to be multiplicative), then e_1 is a singular vector with singular value 1 and all other singular values vanish, so A_p is a rank-one projection. For such A_p one has P_{phi_p} A_p = A_p and hence ||P_{phi_p} A_p||_{S_q}=1, contradicting the required bound >= 1+delta. Consequently, the inference that P_Phi A = tensor_p P_{phi_p} A_p, which in the written proof uses the converse for multiplicative matrices, is not justified for the chosen A_p. The factorization is in fact valid for arbitrary tensor products by an entrywise computation, so the proof can be repaired by removing the multiplicativity assumption and proving the factorization directly, or by choosing A_p with <A_p e_1, e_1>=1 and the ratio property without unit normalization. As written, the key estimate rests on an unproved step.","section":"Section 3, proof of Theorem 2"},{"comment":"The paper applies the norm identity (15) to the infinite tensor products A = tensor_p A_p and tensor_p P_{phi_p} A_p, but it does not verify the hypotheses of [6, Thm. 2.4] quoted in Section 2.3: convergence of product_p ||A_p||, convergence of sum_p ||A_p e_1||-1 and sum_p <A_p e_1,e_1>-1, and membership of the tensor products in S_q. For the specific construction (finitely many finite-rank factors and A_p = H_{e_0} for p > p_N) these conditions hold, so the gap is easily fixed, but the verification should be included because (15) is the mechanism by which the finite-factor lower bound (1+delta)^N is converted into a lower bound for ||P_Phi||.","section":"Section 3, use of identity (15)"}],"minor_comments":[{"comment":"The statement of Theorem 5 claims that there are no bounded projections from the compact (resp. bounded) operators onto the compact (resp. bounded) Helson matrices, but the proof only addresses projections of the weighted form (9). The sentence 'Clearly, a bounded projection P_Phi must satisfy (8)' is justified only for projections of that form, not for arbitrary idempotent maps onto the Helson subspace. Please either restrict the theorem to weighted averaging projections or provide an argument showing that an arbitrary projection can be reduced to this case.","section":"Section 4.1, Theorem 5"},{"comment":"In the final estimates of the two cases in Lemma 4, the interval for x in the infimum is written as 0 <= x <= 1, but the bound involving 1-2x is only meaningful for x <= 1/2 since phi_1 >= 0. It would clarify the argument to state explicitly that x ranges over [0, 1/2] and that the intersection point x_q lies in this interval.","section":"Lemma 4, optimization over x"},{"comment":"The condition <A_p e_1, e_1> = 1 for every p is described as making the tensor product a multiplicative matrix, but the reason is not stated: this condition ensures that the associated function f satisfies f(1,1)=1, which is part of the definition of multiplicativity. A short explanatory sentence would help the reader.","section":"Section 2.3, definition of multiplicative matrix"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct, but the proof of Theorem 2 contains a genuine gap concerning the compatibility of the normalization from Lemma 4 with the multiplicativity assumption. This is repairable by proving the tensor factorization directly for arbitrary tensor products or by adjusting the choice of A_p. The additional result Theorem 5 is currently overstated relative to its proof. I recommend major revision rather than rejection, as the central claim appears defensible after a localized rewrite."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis paper answers a natural question: Peller's averaging projection for Hankel matrices is bounded on S_q for 1<q<∞, while the multiplicative analogue for Helson matrices is not. The authors prove this for the uniform projection (Theorem 1) and for every non-negative multiplicative weighting (Theorem 2), with a uniform lower bound >1 for each q≠2. Lemma 4, the key computation, is solid: the singular values of the test matrices and the subsequent optimizations check out, and the explicit q=1 constant is a nice touch. The additional results on compact/bounded operators and duality are short and correctly stated. The paper is well written and honest in its claims.\n\nThe soft spot is in the proof of Theorem 2 in Section 3. The proof starts with the assumption that A = ⊗_p A_p is a multiplicative matrix, which requires ⟨A_p e1,e1⟩=1 for every p. Then, for p≤p_N, it chooses A_p with ‖A_p‖_{S_q}=1 and ‖P_{φ_p}A_p‖_{S_q}≥1+δ. These two requirements are incompatible. If ‖A_p‖_{S_q}=1 and ⟨A_p e1,e1⟩=1, then e1 is a singular vector with singular value 1 and all other singular values vanish, so A_p=H_{e0}. But then P_{φ_p}A_p=A_p, giving norm 1 rather than ≥1+δ. Thus the witnesses from Lemma 4 cannot be normalized and still satisfy the multiplicative hypothesis.\n\nThe gap is repairable. The factorization P_ΦA = ⊗_p P_{φ_p}A_p does not actually require A to be multiplicative; it holds entrywise for any tensor product because the divisor sum factors over primes. The proof can be fixed by not normalizing A_p and using the ratio ‖P_{φ_p}A_p‖/‖A_p‖≥1+δ directly. The norm identity (15) then gives the desired (1+δ)^N bound. But as written, the proof is missing this step and needs revision.\n\nThe citation pattern is fine: [6] is the external source for the tensor norm identity, and the self-citations are context, not load-bearing. This paper deserves a serious referee; the main result is natural and the fix is straightforward. My recommendation is to send it to peer review and require the tensor-product step to be rewritten before acceptance—not a desk reject.\n\nBest,\n[You]","headline":"The paper gives a clean negative answer to a natural analogue of Peller's theorem, but the proof of the main theorem has a repairable gap in the tensor-product step.","tokens_in":9523,"tokens_out":12607,"would_cite":true,"duration_ms":116388,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B35","47B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The orthogonal projection from Hilbert–Schmidt operators onto Helson matrices, and every non-negative multiplicative weighted averaging projection onto them, is unbounded on the Schatten class $S_q$ for every $q \\neq 2$.","keywords":["Helson matrices","Schatten classes","Hankel matrices","bounded projections","multiplicative functions","infinite tensor products","Dirichlet series","weighted averaging projection"],"falsifier":"Take $q=4$ and a multiplicative weight $\\Phi$ from the paper's family $\\Phi_{\\alpha,\\beta}$, choose $N$ primes, set $A_p$ to the $3 \\times 3$ matrix $C(t)$ from Lemma 4 for $p \\leq p_N$ and $A_p = H_{e_0}$ otherwise, and compute the ratio $\\|P_\\Phi(\\bigotimes_{p \\leq p_N} A_p)\\|_{S_4} / \\|\\bigotimes_{p \\leq p_N} A_p\\|_{S_4}$. If this ratio stays bounded as $N$ grows, Theorem 2 is false; the proof predicts it grows like $(1+\\delta_4)^N$.","tokens_in":8243,"feed_emoji":"🧮","tokens_out":10563,"duration_ms":102188,"temperature":0.7,"pith_summary":"Helson matrices are infinite matrices whose entry in position $(m,n)$ depends only on the product $mn$; they are the multiplicative counterparts of Hankel matrices and appear in the study of Dirichlet series. For Hankel matrices, a classical result says the averaging projection is bounded on every Schatten class $S_q$ for $1<q<\\infty$. This paper proves the opposite in the multiplicative setting: the orthogonal projection from Hilbert–Schmidt operators onto Helson matrices is unbounded on $S_q$ whenever $q \\neq 2$. The main theorem strengthens this to every non-negative multiplicative weight $\\Phi$ satisfying the projection condition, so no such reweighting of the averaging projection can rescue boundedness. Since $S_2$ is the only Schatten class where the projection is a Hilbert-space orthogonal projection, the paper pins down exactly where the Helson analogue of the classical Hankel theorem fails.","feed_headline":"Helson-matrix projection is unbounded for every Schatten q except 2","feed_subtitle":"Unlike the Hankel averaging projection, which is bounded on every S_q, the multiplicative analogue blows up for q≠2.","key_machinery":"The argument runs on two rails. First, $\\ell^2(\\mathbb{N})$ is identified with the infinite tensor product over primes $\\bigotimes_p \\ell^2(\\langle p \\rangle)$, so a multiplicative matrix factors as $A = \\bigotimes_p A_p$ and a multiplicative Helson matrix corresponds to a tensor product of Hankel matrices. Second, for each prime $p$ the weight $\\Phi$ induces an additive weight $\\varphi_p(i,j) = \\Phi(p^i, p^j)$ satisfying $\\sum_{i+j=k} \\varphi_p(i,j) = 1$, and the projection factors as $P_\\Phi(\\bigotimes_p A_p) = \\bigotimes_p P_{\\varphi_p}(A_p)$. Lemma 4 is the quantitative heart: for every $q \\neq 2$ there is a universal $\\delta_q > 0$ such that every weighted Hankel averaging projection $P_\\varphi$ has norm at least $1 + \\delta_q$ on $S_q$, proved by testing on the four $3 \\times 3$ matrices $A(t), B(t), C(t), D(t)$ and optimizing the resulting lower bounds. Feeding the lemma into the tensor-product factorization and using the norm identity $\\|\\bigotimes_p A_p\\|_{S_q} = \\prod_p \\|A_p\\|_{S_q}$ yields $\\|P_\\Phi\\|_{S_q \\to S_q} \\geq (1+\\delta_q)^N$ for every $N$, forcing unboundedness.","core_discovery":"The paper's central claim is Theorem 2: for any non-negative multiplicative function $\\Phi : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{R}$ with $\\sum_{mn=k} \\Phi(m,n) = 1$ for every $k \\geq 1$, the weighted projection $P_\\Phi$ defined by $(P_\\Phi A)_{m,n} = \\varrho_{mn}$ with $\\varrho_k = \\sum_{mn=k} \\Phi(m,n) a_{m,n}$ is unbounded on the Schatten class $S_q$ for every $1 \\leq q \\neq 2 < \\infty$. In particular, the uniform averaging projection $P$ of Theorem 1—the orthogonal projection from $S_2$ onto Hilbert–Schmidt Helson matrices—does not extend to a bounded operator on any other Schatten class. The authors view this as the multiplicative failure of the classical boundedness theorem for Hankel projections, and they also show that there are no bounded projections onto the spaces of compact or bounded Helson matrices, and that the natural duality pairing between Helson matrices in $S_q$ and $S_r$ (with $1/q + 1/r = 1$) is not surjective for $q \\neq 2$.","pith_inferences":["The method suggests that any bounded projection onto Helson matrices on $S_q$ (if one exists for $q \\neq 2$) would have to be non-multiplicative or take negative values; the paper leaves that possibility open.","The proof's lower bound is constructive: for any finite set of primes it builds explicit operators, tensoring the $3 \\times 3$ matrices from Lemma 4, where the projection gains a factor $(1+\\delta)^N$, so the unboundedness is witnessed by finite-dimensional data and could be checked numerically.","Because the obstruction comes from the infinite product structure over primes, analogous projection problems for matrices indexed by other multiplicative semigroups might be attacked with the same two-step argument of tensor factorization plus a uniform additive lower bound."],"forward_implications":["Theorem 1 settles the multiplicative analogue of the classical Hankel projection theorem: the Hilbert–Schmidt orthogonal projection onto Helson matrices has no bounded extension to $S_q$ for any $q \\neq 2$.","Theorem 2 rules out a whole family of repairs: no non-negative multiplicative weight satisfying the projection condition can make a weighted averaging projection onto Helson matrices bounded on $S_q$ for $q \\neq 2$.","There are no bounded projections from the compact operators onto the compact Helson matrices, nor from the bounded operators onto the bounded Helson matrices, even without assuming multiplicativity of the weight.","For $1 < q \\neq 2 < \\infty$ and $1/q + 1/r = 1$, the natural embedding of the Helson matrices in $S_r$ into the dual of the Helson matrices in $S_q$ is not surjective; the duality structure of Helson matrices in Schatten classes differs from that of Hankel matrices.","Because the lower bound $\\|P_\\Phi\\| \\geq (1+\\delta)^N$ holds for every finite set of primes, finite-dimensional tensor-product examples exhibit the unboundedness quantitatively."],"supporting_citations":[{"why":"Supplies the convergence criteria guaranteeing that the infinite tensor product of the operators $A_p$ is a well-defined bounded operator.","marker":"[3]"},{"why":"Supplies the norm identity $\\|\\bigotimes_p A_p\\|_{S_q} = \\prod_p \\|A_p\\|_{S_q}$ that turns the componentwise lower bound into an exponential lower bound for $P_\\Phi$.","marker":"[6]"},{"why":"Provides the classical boundedness of the Hankel averaging projection on $S_q$ that the Helson result is the multiplicative counterpart of.","marker":"[9]"},{"why":"Provides the Hankel projection criteria and the theorems on absence of bounded Hankel projections from compact or bounded operators used in Theorem 5.","marker":"[10]"}],"fun_headline_variants":["Helson projection fails on every Schatten class except 2","Unbounded Helson projections: no Schatten extension for q≠2","Helson matrix projection blows up in every S_q, q≠2","For Helson projections, Schatten q≠2 is unbounded","No bounded projection onto Helson matrices for S_q, q≠2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the identity that the Schatten $q$-norm of an infinite tensor product of operators is the product of their individual Schatten $q$-norms, together with the convergence of the specific infinite tensor products constructed from the $3 \\times 3$ test matrices; if that norm identity or convergence fails for these operators, the exponential lower bound collapses.","fun_headline_variants_meta":{"raw":{"variants":["Helson projection fails on every Schatten class except 2","Unbounded Helson projections: no Schatten extension for q≠2","Helson matrix projection blows up in every S_q, q≠2","For Helson projections, Schatten q≠2 is unbounded","No bounded projection onto Helson matrices for S_q, q≠2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000567,"raw_usage":{"total_tokens":2682,"prompt_tokens":940,"completion_tokens":1742,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":1648}},"tokens_in":556,"tokens_out":1742,"duration_ms":14025,"temperature":1.0,"reasoning_tokens":1648,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:44:10.152147+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $q=4$ and a multiplicative weight $\\Phi$ from the paper's family $\\Phi_{\\alpha,\\beta}$, choose $N$ primes, set $A_p$ to the $3 \\times 3$ matrix $C(t)$ from Lemma 4 for $p \\leq p_N$ and $A_p = H_{e_0}$ otherwise, and compute the ratio $\\|P_\\Phi(\\bigotimes_{p \\leq p_N} A_p)\\|_{S_4} / \\|\\bigotimes_{p \\leq p_N} A_p\\|_{S_4}$. If this ratio stays bounded as $N$ grows, Theorem 2 is false; the proof predicts it grows like $(1+\\delta_4)^N$.","supporting_citations":[{"cited_title":"Guichardet, Tensor products of C∗ -algebras, part II: Infinite tensor products , Aarhus Universitet Lecture Notes Series, no","cited_arxiv_id":null,"evidence_quote":"Supplies the convergence criteria guaranteeing that the infinite tensor product of the operators $A_p$ is a well-defined bounded operator."},{"cited_title":"532 (2017), 179–197","cited_arxiv_id":null,"evidence_quote":"Supplies the norm identity $\\|\\bigotimes_p A_p\\|_{S_q} = \\prod_p \\|A_p\\|_{S_q}$ that turns the componentwise lower bound into an exponential lower bound for $P_\\Phi$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical boundedness of the Hankel averaging projection on $S_q$ that the Helson result is the multiplicative counterpart of."},{"cited_title":"Peller, Hankel operators and their applications , Springer Monographs in Mathe- matics, Springer-Verlag, New York, 2003","cited_arxiv_id":null,"evidence_quote":"Provides the Hankel projection criteria and the theorems on absence of bounded Hankel projections from compact or bounded operators used in Theorem 5."}],"review_version":1}