{"id":"d0e44758-0b67-4733-b5d9-3417fddcae85","arxiv_id":"2412.00482","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a sign-based variant of Delsarte and Turán problems, extremal functions exist on LCA groups when the allowed sets are boundary-coherent.","lead":"The paper proves that a maximizing function exists for a new variant of Delsarte and Turán extremal problems on locally compact abelian groups, under a mild topological condition called boundary-coherence. This makes a broad family of linear programming bounds in harmonic analysis well-posed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed boundary-coherence definition (∂Ω ⊂ ext Ω) contradicts the proof, which uses ∂Ω ⊂ ∂(ext Ω); as stated, Theorem 17 covers only clopen sets, so the main existence theorem needs a corrected definition.","rationale":"The reader's weakest_assumption already identifies the boundary-coherence definition as the load-bearing premise, and I agree. The proof of Theorem 14 is structurally sound under the intended condition; the only use of the hypothesis is to upgrade the a.e. inclusions S± \\ Ω± to full inclusions. The printed definition would make the hypothesis almost never satisfied unless Ω± are clopen, so the theorem as stated does not deliver the advertised generalization to open sets such as intervals or convex bodies. The defect is repairable by replacing Definition 1 with ∂Ω ⊂ ∂(ext Ω), and the paper's own Lemma 16 and surrounding text confirm this was the intent. I also note a separate internal inconsistency in the definition of support (supp f := {f ≠ 0}), which would make F* = F and void Section 5; however, this does not affect the proof of Theorem 17 and is a further reason the paper needs line edits. Because the main theorem is correct under the corrected definition, the conditional verdict stands.","tokens_in":16288,"tokens_out":22329,"duration_ms":204804,"concrete_test":"Check Ω = (-1,1) in R: compute ∂Ω = {-1,1}, ext Ω = R\\[-1,1], cl(ext Ω) = (-∞,-1] ∪ [1,∞). The printed condition ∂Ω ⊂ ext Ω fails (∂Ω ∩ ext Ω = ∅), while the intended condition ∂Ω ⊂ ∂(ext Ω) holds. Then inspect the final paragraph of the proof of Theorem 14: the argument 'there are points z ∈ V ∩ ext Ω+' requires x ∈ cl(ext Ω+), which is exactly the intended condition. If the test confirms this mismatch, the theorem's hypothesis must be read with the corrected definition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 1 defines boundary-coherence as ∂Ω ⊂ ext Ω, where ext Ω = X \\ cl(Ω). Since ext Ω and ∂Ω are disjoint by construction, the literal condition forces ∂Ω = ∅, i.e., Ω is clopen. The paragraph following Definition 1 claims equivalence with ∂Ω ⊂ ∂(ext Ω), but that equivalence is false: for Ω = (-1,1) ⊂ R, ∂Ω = {-1,1} satisfies ∂Ω ⊂ ∂(ext Ω) but not ∂Ω ⊂ ext Ω, because ∂Ω ∩ ext Ω = ∅. The proof of Theorem 14 invokes boundary-coherence exactly once, in the final sign-inclusion step, to ensure that every x ∉ Ω+ has a neighbourhood intersecting ext Ω+; this requires x ∈ cl(ext Ω+), which is exactly the intended condition ∂Ω+ ⊂ ∂(ext Ω+). Thus, as printed, the theorem is technically true only for clopen sets and fails to cover the motivating open convex sets and intervals. A secondary defect is the unproved assertion C_G(Ω+,Ω-) > 0 in the same proof; this follows from Lemma 12 but is not stated explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a general extremal problem for continuous positive definite functions on locally compact Abelian groups, where the usual support conditions are replaced by the weaker preimage conditions f^{-1}(0,∞) ⊂ Ω_+ and f^{-1}(-∞,0) ⊂ Ω_-. It introduces the corresponding Delsarte and Turán variants, proves reductions to σ-compact open subgroups, and establishes an existence theorem for extremal functions under a new topological condition called \"boundary-coherence\" (Theorem 17). The paper also compares the new formulation with the original support-based one, proving equivalence of the extremal constants for boundary-coherent sets and for star-shaped sets in topological vector spaces, with several examples illustrating sharpness.","tokens_in":16502,"tokens_out":7637,"duration_ms":73622,"significance":"If the main existence theorem is correct, it provides a broad existence result for Delsarte- and Turán-type extremal problems on general LCA groups under an explicit topological hypothesis, substantially extending previous work restricted to specific groups or sets. The paper is careful in reducing to σ-compact groups, uses standard tools (weak compactness, Mazur's lemma, approximation of unity) in a transparent way, and includes instructive examples showing that boundary-coherence is not necessary for existence. The equivalence section also clarifies the relationship between the two formulations. However, as printed, the central definition is misstated, and the proof of the main theorem relies on a different, intended definition; this must be repaired before the results can be considered reliable.","major_comments":[{"comment":"Definition 1 defines boundary-coherence as ∂Ω ⊂ ext Ω, where ext Ω = X \\ cl(Ω). Since ∂Ω and ext Ω are disjoint by construction, this condition forces ∂Ω = ∅, i.e., Ω is clopen. The paragraph after Definition 1 claims equivalence with ∂Ω ⊂ ∂(ext Ω), but this equivalence is false: for Ω = (-1,1) ⊂ R, ∂Ω = {-1,1} satisfies ∂Ω ⊂ ∂(ext Ω) but does not satisfy ∂Ω ⊂ ext Ω. The proof of Theorem 14 uses the intended condition: for arbitrary x ∉ Ω_+ it invokes \"by boundary-coherence there are points z ∈ V ∩ ext Ω_+\", which requires x ∈ cl(ext Ω_+), i.e., ∂Ω_+ ⊂ ∂(ext Ω_+). Consequently, as printed, Theorem 17 covers only clopen sets and does not cover the motivating open convex sets, intervals, or Examples 21/28. The definition and the paragraph after it must be corrected to ∂Ω ⊂ ∂(ext Ω), and all statements and proofs (Theorems 14 and 17, Corollaries 19 and 20, Proposition 24, Lemma 16) should be read and restated with this intended notion.","section":"Definition 1; proof of Theorem 14; Theorem 17"},{"comment":"The proof asserts \"The fact that ∫_G f dλ_G ≥ C_G(Ω_+,Ω_-) > 0\" without justification. Strict positivity of C_G(Ω_+,Ω_-) is not automatic from the definition of the extremal constant and is used to deduce f(0) = 1. The inequality is true and can be proved from Lemma 12 by choosing a compact symmetric neighbourhood K of 0 with K+K ⊂ Ω_+ and considering a normalized convolution f = (1_K * 1_K)/λ(K), which lies in F_G(Ω_+,Ω_-) and has positive integral λ(K); the argument should be supplied in the text rather than left implicit.","section":"Section 4, proof of Theorem 14"},{"comment":"The sentence \"Firstly, the equality F (Ω_+, Ω_-) = F*(Ω_+, Ω_-) is easily seen to be true, always\" is false, and the displayed chain (18) ends with the same equality. Example 22 gives a direct counterexample: for Ω = (-1,1) ⊂ R, the triangular extremal function lies in F_R(Ω,Ω) but not in F*_R(Ω,Ω). This is a substantive error in the equivalence discussion, although it appears to be a local slip; the chain and the surrounding claims need to be corrected, for instance by removing the final equality and stating it only under the additional assumptions of Proposition 24 or Corollary 26 where it actually holds.","section":"Section 5, Eq. (18) and preceding paragraph"}],"minor_comments":[{"comment":"Example 21 and Example 28 are identical (same set Ω, same extremal constant, same figures); one of them should be removed or the two occurrences should be cross-referenced instead of repeated.","section":"Examples 21 and 28"},{"comment":"The line \"∫_G f_− λ_G dλ_G ≤ lim inf ...\" contains a misplaced λ_G; it should read ∫_G f_− dλ_G ≤ lim inf ... . The same typo appears in the inequality two lines above (15).","section":"Proposition 15, display after (14)"},{"comment":"The notation in the chains (18) and (19) is ambiguous because closure bars appear to be missing (e.g., terms like F (int Ω_+, int Ω_-) are not clearly distinguished from F (overline{int Ω_+}, overline{int Ω_-})). The typesetting should be cleaned so that each equality is unambiguously stated.","section":"Display (18) and (19)"},{"comment":"The statement of Lemma 16 says \"i.e., ∂_H(Ω ∩ H) ⊂ ext_H(Ω ∩ H)^H\", which is the printed (literal) form of boundary-coherence; once Definition 1 is corrected, the statement should be updated to the intended form ∂_H(Ω ∩ H) ⊂ ∂_H(ext_H(Ω ∩ H)) to match the proof, which uses the corrected notion.","section":"Lemma 16"}],"recommendation":"major_revision","confidential_remarks":"The central existence proof is fundamentally sound under the intended boundary-coherence condition, and the misstatement of Definition 1 is repairable by a local correction. I am nonetheless recommending major revision rather than minor because the printed definition makes the main theorem false as stated and because the equivalence section contains a false assertion contradicted by one of the paper's own examples. With the definition corrected and the cited gaps filled, the paper would be a solid contribution. The repeated Example 21/28 and the typesetting issues in Section 5 suggest a need for a careful pass before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is the sign-set formulation (f^{-1}(0,∞), f^{-1}(-∞,0) constraints) rather than the usual support formulation, plus an existence theorem for this variant under a boundary-coherence assumption. The paper does this carefully: it reduces to σ-compact groups, proves equivalence between the two formulations under the intended topological condition, and gives clean corollaries for Delsarte and Turán problems. The credit to Mavroudis for Theorem 32 is honest, and the self-citations are to published work with independent proofs, so no circularity problem there.\n\nThe main proof strategy is genuinely good: weak compactness in L2, Mazur's lemma to get strong convergence, then a careful sign-inclusion argument to show the limit lies in the desired class. The authors also admit openly that boundary-coherence might be unnecessary and that they could not remove it; Example 21 shows that extremal functions can exist without it. That is a fair limitation statement.\n\nNow the soft spots. The printed Definition 1 says a set is boundary-coherent if ∂Ω ⊂ ext Ω, where ext Ω is the complement of the closure. That forces ∂Ω = ∅, so only clopen sets satisfy it. The surrounding text says this is equivalent to ∂Ω ⊂ ∂(ext Ω), but that equivalence is false (take Ω = (-1,1) in R). Every proof in the paper uses the second property, which is the intended one. So as printed, Theorem 17 is vacuous for any set with nonempty boundary, which includes the motivating open convex sets. This is a real error but a trivial one: change Definition 1 to ∂Ω ⊂ ∂(ext Ω) and nothing else in the paper needs adjusting. A referee should insist on that fix but not treat it as a deep flaw.\n\nA minor issue: in the proof of Theorem 14, the assertion C_G(Ω+,Ω−) > 0 is used without proof. It follows from Lemma 12 by taking an appropriate positive definite kernel supported near 0, but it should be stated explicitly.\n\nThe equivalence section has some redundancy, and the containment (18)-(19) is a bit dense, but the results are correct. The paper is written for people working on positive definite functions, Delsarte-type extremal problems, and packing on LCA groups. I would send it to a serious referee with a request to fix Definition 1 and add the C_G positivity note. The mathematics is solid, the contribution is new, and the exposition is mostly clear. My verdict: accept after minor revision.","headline":"Solid existence theorem for a new sign-set Delsarte/Turán variant on LCA groups, but the printed boundary-coherence definition is self-contradictory; the fix is one line and the proofs already use the right condition.","tokens_in":17070,"tokens_out":1857,"would_cite":true,"duration_ms":21131,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["43A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Boundary-coherent sign sets always admit extremal functions for Delsarte- and Turán-type problems.","keywords":["locally compact Abelian groups","positive definite functions","Delsarte extremal problem","Turán extremal problem","existence of extremal functions","boundary-coherent sets","sign-set formulation"],"falsifier":"Find a locally compact Abelian group $G$ and symmetric boundary-coherent sets $\\Omega_+$, $\\Omega_-$, with $\\Omega_+$ a neighbourhood of 0 having finite Haar measure, for which the supremum defining $C_G(\\Omega_+,\\Omega_-)$ is not attained by any function in $F_G(\\Omega_+,\\Omega_-)$. Theorem 17 asserts no such pair exists, so one concrete pair would settle the question.","tokens_in":16083,"feed_emoji":"📐","tokens_out":9655,"duration_ms":84583,"temperature":0.7,"pith_summary":"The paper sets up a common umbrella for Delsarte- and Turán-type extremal problems on locally compact Abelian groups, using the weaker condition that the positivity and negativity sets of a function are contained in prescribed sets, instead of requiring containment of supports. Its main theorem states that if those prescribed sets are boundary-coherent, symmetric, and the positive set is a neighbourhood of zero with finite Haar measure, then the extremal value is actually attained by an admissible function. This is an existence result, not just an estimate: it turns the supremum defining the problem into a maximum. The authors also show that for boundary-coherent sets this new formulation is equivalent to the original support-based formulation, so the existence conclusion carries over to the classical Delsarte and Turán constants.","feed_headline":"Boundary-coherent sets guarantee extremal functions","feed_subtitle":"A mild topological condition on the sign sets forces a maximizing positive definite function to exist on any LCA group.","key_machinery":"The central condition is boundary-coherence: a set $\\Omega$ is boundary-coherent when every boundary point is also a boundary point of its exterior $\\operatorname{ext}\\Omega = X\\setminus\\overline{\\Omega}$, so boundary points can be approached from outside the closure. As printed, Definition 1 states $\\partial\\Omega\\subset\\operatorname{ext}\\Omega$, but the intended and used condition is $\\partial\\Omega\\subset\\partial(\\operatorname{ext}\\Omega)$. The proof machinery is a weak-compactness argument: extremal sequences are bounded in $L^2(G)$ by the finite measure of $\\Omega_+$; the convex-combination lemma turns the weak limit into a strong $L^2$ and almost-everywhere limit of convex combinations; the limit is integrally positive definite and hence, by a standard theorem on such functions, agrees almost everywhere with a continuous positive definite function; boundary-coherence then forces the sign preimages of the corrected limit into $\\Omega_+$ and $\\Omega_-$. This last step is where the topological hypothesis does its work.","core_discovery":"On any locally compact Abelian group $G$ with Haar measure $\\lambda_G$, the paper proves that the extremal problem $C_G(\\Omega_+,\\Omega_-)=\\sup\\{\\int_G f\\,d\\lambda_G : f\\in F_G(\\Omega_+,\\Omega_-)\\}$, where $F_G(\\Omega_+,\\Omega_-)$ consists of continuous positive definite functions with $f(0)=1$, integrable, and satisfying $f^{-1}(0,\\infty)\\subset\\Omega_+$ and $f^{-1}(-\\infty,0)\\subset\\Omega_-$, has a solution whenever $\\Omega_+$ and $\\Omega_-$ are symmetric, boundary-coherent, and $\\Omega_+$ is a neighbourhood of 0 of finite Haar measure. The proof first reduces the general group case to a $\\sigma$-compact open subgroup, constructs an extremal function there by weak compactness in $L^2$ combined with a convex-combination lemma and a standard lifting of integrally positive definite functions to continuous ones, and then extends it by trivial extension. Boundary-coherence is used precisely once, to force the sign sets of the limiting function to lie in the prescribed sets. Corollaries assert the same existence for the Delsarte constant $D_G(\\Omega)$ and the Turán constant $T_G(\\Omega)$ when the defining set is boundary-coherent.","pith_inferences":["The boundary-coherence assumption is probably not necessary for existence: Example 21 gives a non-coherent set with an extremal function, so the exact class of sets guaranteeing existence is still open.","Because the main theorem reduces to $\\sigma$-compact subgroups, the result applies uniformly to finite-dimensional, infinite-dimensional, and totally disconnected LCA groups; the hypothesis is purely topological and independent of the group's lattice structure.","The equivalence results suggest that in future applications of Delsarte-type bounds one may freely switch between support and preimage formulations whenever the defining sets are boundary-coherent, which could simplify the search for extremal functions in concrete packing and energy problems."],"forward_implications":["The Delsarte and Turán constants are attained for every boundary-coherent symmetric neighbourhood of 0 with finite Haar measure, on any LCA group (Corollaries 19 and 20).","For boundary-coherent sets, the new sign-set class equals the original support-based class, so the two formulations of the extremal problem have the same admissible functions and the same constants (Proposition 24 and Corollary 27).","In topological vector spaces over $\\mathbb{R}$, for bounded 0-symmetric sets satisfying a radial-containment condition, all six extremal constants in the comparison chain coincide (Corollary 34).","Every extremal function can be approximated by compactly supported admissible functions in $L^1$ and uniformly on compact sets, so compactly supported test functions are dense enough to compute the constant (Proposition 18)."],"supporting_citations":[{"why":"Introduces the original sign-set extremal setup and supplies the compact-support approximation lemma (Lemma 31) used in the equivalence chain.","marker":"[2]"},{"why":"Supplies the positivity-preserving approximation of unity (Lemma 12) and the Turán zero-set example that anchors the boundary-coherence discussion.","marker":"[13]"},{"why":"Is the earlier existence result for the Delsarte problem that the present theorem extends to the sign-set formulation and to general LCA groups.","marker":"[15]"},{"why":"Provides the band-limited existence result whose restriction the present work drops.","marker":"[10]"},{"why":"Supplies the convex-combination lemma and weak sequential compactness, the mechanism converting weak $L^2$ limits into strong limits.","marker":"[3]"},{"why":"Gives the theorem that integrally positive definite functions agree locally almost everywhere with continuous positive definite functions, used to make the weak limit admissible.","marker":"[17]"},{"why":"Supplies standard facts on $\\sigma$-compact open subgroups and supports of integrable functions used in the reduction theorems.","marker":"[6]"}],"fun_headline_variants":["Boundary coherence yields extremal functions on LCA groups","LCA groups: boundary-coherent sets force extremal solutions","A mild topological condition ensures extremal functions","Boundary-coherent sign sets force extremal functions to exist","Existence of Delsarte extremals from boundary-coherent sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that both $\\Omega_+$ and $\\Omega_-$ satisfy the intended boundary-coherence condition — every boundary point of the set is also a boundary point of its exterior, so boundary points can be approached from outside the closure — and the paper notes this condition is used exactly once, in the final step, and could not be removed.","fun_headline_variants_meta":{"raw":{"variants":["Boundary coherence yields extremal functions on LCA groups","LCA groups: boundary-coherent sets force extremal solutions","A mild topological condition ensures extremal functions","Boundary-coherent sign sets force extremal functions to exist","Existence of Delsarte extremals from boundary-coherent sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000855,"raw_usage":{"total_tokens":3680,"prompt_tokens":875,"completion_tokens":2805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":2723}},"tokens_in":491,"tokens_out":2805,"duration_ms":17546,"temperature":1.0,"reasoning_tokens":2723,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:22:55.388903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a locally compact Abelian group $G$ and symmetric boundary-coherent sets $\\Omega_+$, $\\Omega_-$, with $\\Omega_+$ a neighbourhood of 0 having finite Haar measure, for which the supremum defining $C_G(\\Omega_+,\\Omega_-)$ is not attained by any function in $F_G(\\Omega_+,\\Omega_-)$. Theorem 17 asserts no such pair exists, so one concrete pair would settle the question.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the original sign-set extremal setup and supplies the compact-support approximation lemma (Lemma 31) used in the equivalence chain."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the positivity-preserving approximation of unity (Lemma 12) and the Turán zero-set example that anchors the boundary-coherence discussion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the earlier existence result for the Delsarte problem that the present theorem extends to the sign-set formulation and to general LCA groups."},{"cited_title":"Gaál and Zs","cited_arxiv_id":null,"evidence_quote":"Provides the band-limited existence result whose restriction the present work drops."},{"cited_title":"Functional Analysis, Sobolev Spaces, and Pa rtial Differential Equations","cited_arxiv_id":null,"evidence_quote":"Supplies the convex-combination lemma and weak sequential compactness, the mechanism converting weak $L^2$ limits into strong limits."},{"cited_title":"Positive definite and definitizable functi ons","cited_arxiv_id":null,"evidence_quote":"Gives the theorem that integrally positive definite functions agree locally almost everywhere with continuous positive definite functions, used to make the weak limit admissible."},{"cited_title":"Principles of Harmonic Analy sis","cited_arxiv_id":null,"evidence_quote":"Supplies standard facts on $\\sigma$-compact open subgroups and supports of integrable functions used in the reduction theorems."}],"review_version":1}