{"id":"f5d84768-960c-4011-b797-7f40b79d647c","arxiv_id":"2607.06530","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"The (IR) property—rigidity of independent functions against low-dimensional approximation—is established for broad classes of symmetric spaces (Lorentz L_{p,q} with 1<p<2, 1≤q≤2; Orlicz L_{log^α L} with α≥1/2) and shown to fail when the fundamental function grows faster than √t or when ℓ_q (q>2) isl","lead":"The paper proves that in many symmetric function spaces—including Lorentz and Orlicz spaces—sets of independent mean-zero functions cannot be well approximated by low-dimensional subspaces, and characterizes when this rigidity fails. This extends the known L_p theory to broader classes of spaces, connecting approximation properties to lattice structure.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The proofs of Theorems 1–5 are careful, use well-established tools (Johnson-Schechtman inequalities, Kruglov property, Kolmogorov width estimates), and the key duality and disjointification steps check out under scrutiny.","rationale":"The reader's verdict of ACCEPT with HIGH confidence is appropriate. The paper provides rigorous proofs of both positive results (Theorems 1, 4) and negative results (Theorems 2, 3, 5), with concrete applications to Lorentz and Orlicz spaces. The proofs use established tools correctly: the Johnson-Schechtman/Kruglov inequality (6), the duality between lower and upper 2-estimates, Kolmogorov width estimates for Euclidean balls and octahedra, and Lemma 2.1 from [31]. The key technical steps I scrutinized—the biorthogonal system construction in Theorem 1, the probabilistic construction in Theorem 2, and the disjointification argument in Theorem 5—all hold up under careful examination. The reader correctly identified the Kruglov property as the main structural assumption, and the paper itself acknowledges this (Remark 3.8, Problem 3.14). The open problem L_{2,q} for q<2 is honestly stated. No red flags, no internal inconsistencies, no gaps in the arguments. The novelty is moderate (extending known L_p results to symmetric spaces) but the execution is thorough.","tokens_in":22433,"tokens_out":9886,"duration_ms":174440,"concrete_test":"Verify the claim in the proof of Theorem 5 that the Z²_X-norm of the disjoint copy Σa_k f̄_k reduces to ‖Σa_k x_k‖_X when the x_k are disjoint on [0,1]. Specifically, confirm that (Σa_k f̄_k)* is supported on [0,1] (since total support measure ≤ 1), making the L²[1,∞)-component vanish. This can be checked by direct computation of the decreasing rearrangement for a concrete example, e.g., x_k = χ_{[(k-1)/N, k/N]} in L_p with p>2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After careful reading, I could not identify a load-bearing concern that lands. I checked several potential weak points:\n\n(1) Theorem 1's duality argument: X separable (from lower 2-estimate excluding ℓ∞) → X*=X'; lower 2-estimate on X → upper 2-estimate on X' [27, Prop 1.f.5]; X⊃L₂ → X'⊂L₂. These are standard and correct. The biorthogonal system construction via conditional expectations is valid: g_k = E_{A_k}g'_k - Eg'_k are independent (since σ-algebras A_k are independent), mean zero, and satisfy ∫f_j g_k = δ_{jk}. The ℓ₂-estimate (7) applies to X' because X'∈K, X' has upper 2-estimate, and X'⊂L₂. Lemma 2.1 then yields the averaged width bound. All steps are sound.\n\n(2) Theorem 2's construction: The i.i.d. functions f_k = ±M (prob ε/2 each) or 0 (prob 1-ε) satisfy ‖f_k‖_X = Mφ_X(ε) = 1 and E[f_k]=0. The key step requiring C₁(γ)ε/φ_X(ε)² ≤ 1/2 is ensured by the hypothesis φ_X(t)t^{-1/2}→∞. The tail bound m(C) ≤ 2^{-s*} with s*→∞ as N→∞ gives ‖(f_k-g_k)χ_C‖_X → 0. The proof correctly establishes the direct negation of (IR): for every γ>0, there exist N and f_1,...,f_N with d_{γN} < γ.\n\n(3) Theorem 5's disjointification: Since the disjoint x_k on [0,1] have total support measure ≤1, their disjoint copies on [0,∞) have rearrangement supported on [0,1], so the L²-part of the Z²_X norm vanishes and ‖Σa_k f̄_k‖_{Z²_X} = ‖Σa_k x_k‖_X. Inequality (6) then gives ‖Σa_k f_k‖_X ≤ C(X)‖Σa_k x_k‖_X, combined with (4) yields the ℓ_q-bound, and width estimate (11) for B₁^N in ℓ_q^N finishes the proof.\n\nThe reader's identified concern (Kruglov property not being necessary) is explicitly addressed in Remark 3.8, which shows (IR) can hold without X'∈K (e.g., ExpL_p for 1<p≤2 via [9, Theorem II.9]). The open case L_{2,q} (q<2) is honestly stated in Problem 3.14.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper introduces the (IR) property for symmetric function spaces on [0,1], which captures the rigidity of finite sets of mean-zero independent functions against approximation by low-dimensional linear subspaces. The authors establish broad sufficient conditions for (IR) (Theorem 1, Theorem 4) based on lower 2-estimates, the embedding X⊃L₂, and the Kruglov property of the associate space X'. They also prove failure of (IR) under two types of conditions: when the fundamental function grows faster than t^{1/2} (Theorem 2), and when ℓ_q for q>2 is roughly lattice finitely representable in a space with the Kruglov property (Theorem 5). These results yield a near-complete classification for Lorentz L_{p,q} spaces and partial results for Orlicz spaces. The proofs use Johnson–Schechtman disjointification inequalities, Kruglov property machinery, Kolmogorov width estimates for finite-dimensional bodies, and a duality argument via biorthogonal systems constructed from conditional expectations.","tokens_in":23450,"tokens_out":1727,"duration_ms":351429,"significance":"The paper extends the L_p rigidity theory to the broader setting of symmetric spaces, which is a natural and substantive generalization. The positive results (Theorems 1 and 4) provide a clean, checkable framework: lower 2-estimate plus X⊃L₂ plus X'∈K suffices for (IR), with the key inequality (16) serving as the operative condition. The negative results (Theorems 2 and 5) are complementary and use explicit constructions. The Lorentz classification (Corollaries 3.9, 3.12) is clean, and the Orlicz results (Corollaries 3.7, 3.15, 3.16) cover natural scales. The open problems (3.11, 3.14, 3.18) are well-posed and indicate genuine gaps rather than omissions. The paper builds on the authors' prior work [29, 30, 31] but the extension to symmetric spaces via the Kruglov property and lattice-geometric conditions is new. The proofs are detailed and verifiable; the key steps (separability argument, conditional-expectation biorthogonal construction, application of inequality (7), width bounds) are all carried out explicitly.","major_comments":[{"comment":"Theorem 1 (stated in the Introduction) requires X⊃L₂, but the proof in §3.1 uses this assumption to conclude X'⊂L₂, which is then needed for inequality (7) via [9, Theorem II.2.4]. However, the more general Theorem 4 in §3.1 does not list X⊃L₂ among its hypotheses (condition (i) does, but the main statement only requires inequality (16)). The relationship between Theorem 1 and Theorem 4 should be clarified: is Theorem 1 a strict corollary of Theorem 4(i), or does the proof of Theorem 1 use X⊃L₂ in an essential way beyond what Theorem 4 captures? Remark 3.2 suggests that (16) itself implies X⊃L₂, which would close the loop, but this logical relationship is not stated explicitly in the theorems. Adding a sentence clarifying the precise logical dependencies would strengthen the presentation.","section":null},{"comment":"In the proof of Theorem 2 (§3.2), the construction fixes ε satisfying C₁(γ)ε/φ_X(ε)² ≤ 1/2 and then fixes M via (22). The parameter s* is then set by (24) as s* = ⌊Nγ²/(4C(γ)²M²)⌋. For the tail bound m(C) ≤ 2^{-s*} to yield Mφ_X(2^{-s*}) ≤ γ/2, one needs s* → ∞ as N → ∞, which requires M to be fixed (independent of N). Since M = 1/φ_X(ε) and ε is fixed, this is indeed the case. However, the argument would benefit from an explicit statement that M is a constant depending only on X and γ (not on N), as this is load-bearing for the limit s* → ∞.","section":null},{"comment":"Proposition 3.19 (§3.3) extends the (IR)-type lower bound to functions satisfying a support condition measured by R_{1-δ}(f). The proof involves a dilation argument with factor R (line after equation (34)) that uses the bound ‖σ̃_R‖ ≤ max{1, R} on symmetric spaces on the half-axis. The application of this dilation to the Z²_{X'} norm is correct, but the step from (34) to the final estimate involves several implicit constants (C_X from the lower 2-estimate, δ^{-1}, R, C(X)). It would help the reader to state explicitly that the final constant B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2) is the one appearing in the proposition's conclusion, and to confirm that all dependencies are as claimed (B depends on X, δ, and R_{1-δ}(f)).","section":null}],"minor_comments":[{"comment":"Abstract: 'sufficienly' should read 'sufficiently'.","section":null},{"comment":"Proof of Theorem 2 (§3.2): 'Conbining' should read 'Combining'.","section":null},{"comment":"§2.1: 'detailes' should read 'details'.","section":null},{"comment":"§2.2: 'duscussed' should read 'discussed'.","section":null},{"comment":"Reference [16]: the page range contains a formatting artifact ('1048ı¿œ1052'); please correct to '1048–1052'.","section":null},{"comment":"Reference [21]: the author list formatting ('A. Kamińska A., L. Maligranda, L.-E. Persson') appears to have a stray 'A.' after the first author's name; please verify and correct.","section":null},{"comment":"In the proof of Theorem 1 (§3.1), the step verifying ∫f_j g_k = 0 for j≠k uses independence of f_j and E_{A_k}g'_k. It would improve readability to note explicitly that this independence follows from the independence of the σ-algebras A_j and A_k for j≠k, which is a standard but non-trivial fact.","section":null},{"comment":"In Proposition 3.19 (§3.3), the quantity R_{1-δ}(f) is defined as a sum over k of m{|f_k| > 1-δ}, but it is used in the proof as a single number R. A brief remark that R_{1-δ}(f) is finite by the assumption R_0(f) < ∞ (in Corollary 3.20) or by the finiteness of the collection would help the reader.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution to the rigidity theory of independent functions. The proofs are correct and the results are new. The major comments above are primarily about clarity of logical dependencies and explicitness of constants, not about correctness of the core arguments. I recommend minor revision. The self-citation pattern ([29, 30, 31]) is appropriate given that the (IR) property was introduced in those works; the present paper substantially extends the framework. The open problem about L_{2,q} for q<2 (Problem 3.14) is genuine: the space satisfies a lower 2-estimate but does not contain L₂, so neither Theorem 1 nor Theorem 5 applies."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for three constructive comments, all of which concern clarification of logical dependencies and explicit tracking of constants. We agree with all three points and will incorporate the requested clarifications in the revised manuscript.","responses":[{"response":"The referee is correct that the logical dependencies between Theorem 1, Theorem 4, and Remark 3.2 are not stated as explicitly as they should be. To clarify: Theorem 1 is indeed a corollary of Theorem 4(i). The proof of Theorem 1 in §3.1 proceeds by verifying the hypotheses of Theorem 4(i): the lower 2-estimate and X⊃L₂ together imply (by duality) that X' satisfies an upper 2-estimate and X'⊂L₂; combined with X'∈K, these yield inequality (16) via [9, Theorem II.2.4]. Thus X⊃L₂ is used in the proof of Theorem 1 only to establish (16), not in any additional way. Furthermore, as the referee notes, Remark 3.2 (citing [7, Theorem 38(b)]) shows that (16) itself implies both a lower 2-estimate and X⊃L₂, so condition (16) is in fact equivalent to the conjunction of these properties (given X'∈K). We will add an explicit sentence after the statement of Theorem 4 clarifying that Theorem 1 is a direct corollary of Theorem 4(i), and we will cross-reference Remark 3.2 to note that (16) implies X⊃L₂, closing the logical loop.","revision_made":"yes","referee_comment":"Theorem 1 requires X⊃L₂, but the proof in §3.1 uses this to conclude X'⊂L₂, needed for inequality (7) via [9, Theorem II.2.4]. The more general Theorem 4 does not list X⊃L₂ among its hypotheses (condition (i) does, but the main statement only requires inequality (16)). The relationship between Theorem 1 and Theorem 4 should be clarified: is Theorem 1 a strict corollary of Theorem 4(i), or does the proof of Theorem 1 use X⊃L₂ in an essential way beyond what Theorem 4 captures? Remark 3.2 suggests that (16) itself implies X⊃L₂, which would close the loop, but this logical relationship is not stated explicitly in the theorems."},{"response":"The referee's observation is correct. The parameter ε is fixed depending only on X and γ (via the condition C₁(γ)ε/φ_X(ε)² ≤ 1/2), and M = 1/φ_X(ε) is then also fixed depending only on X and γ. Consequently s* = ⌊Nγ²/(4C(γ)²M²)⌋ → ∞ as N → ∞, which is what drives the tail estimate Mφ_X(2^{-s*}) → 0. We will add an explicit sentence after the point where ε and M are fixed, stating that both ε and M are constants depending only on X and γ (and not on N), and that this independence from N is what ensures s* → ∞ as N → ∞.","revision_made":"yes","referee_comment":"In the proof of Theorem 2 (§3.2), the construction fixes ε satisfying C₁(γ)ε/φ_X(ε)² ≤ 1/2 and then fixes M via (22). The parameter s* is set by (24) as s* = ⌊Nγ²/(4C(γ)²M²)⌋. For the tail bound m(C) ≤ 2^{-s*} to yield Mφ_X(2^{-s*}) ≤ γ/2, one needs s* → ∞ as N → ∞, which requires M to be fixed (independent of N). Since M = 1/φ_X(ε) and ε is fixed, this is indeed the case. However, the argument would benefit from an explicit statement that M is a constant depending only on X and γ (not on N), as this is load-bearing for the limit s* → ∞."},{"response":"We agree that the tracking of constants in the proof of Proposition 3.19 could be made more transparent. The final constant is indeed B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2), where C(X) is the constant from the disjointification inequality (6)/(33), C_X is the constant from the lower 2-estimate of X (equivalently, the upper 2-estimate of X'), δ is the parameter from the support condition, and R = R_{1-δ}(f) = Σ_k m{|f_k| > 1-δ}. All dependencies are as claimed: B depends on X (through C(X) and C_X), on δ, and on R_{1-δ}(f). We will add an explicit display of the constant B at the end of the proof, immediately before the concluding sentence, and confirm the dependency structure.","revision_made":"yes","referee_comment":"Proposition 3.19 (§3.3) extends the (IR)-type lower bound to functions satisfying a support condition measured by R_{1-δ}(f). The proof involves a dilation argument with factor R (line after equation (34)) that uses the bound ‖σ̃_R‖ ≤ max{1, R} on symmetric spaces on the half-axis. The application of this dilation to the Z²_{X'} norm is correct, but the step from (34) to the final estimate involves several implicit constants (C_X from the lower 2-estimate, δ^{-1}, R, C(X)). It would help the reader to state explicitly that the final constant B = δ^{-1}C(X)R_{1-δ}(f)(C_X + 2) is the one appearing in the proposition's conclusion, and to confirm that all dependencies are as claimed (B depends on X, δ, and R_{1-δ}(f))."}],"tokens_in":22649,"tokens_out":1370,"duration_ms":192573,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper extends the (IR) property — rigidity of sets of independent mean-zero functions against low-dimensional linear approximation — from the L_p scale to general symmetric spaces. The main results are clean: Theorem 1 gives sufficient conditions (lower 2-estimate, X⊃L₂, X' has Kruglov property), Theorems 2 and 3 give obstructions (fundamental function growing faster than √t, or ℓ_q roughly lattice finitely representable with q>2), and these combine to give a complete classification for Lorentz L_{p,q} spaces: (IR) holds for 1<p<2, 1≤q≤2, and fails for max{p,q}>2. The Orlicz case is partially covered with explicit conditions on Φ. This is a genuine extension of prior work by Malykhin, and the results are natural and well-motivated. The proofs are careful and use established tools — Johnson-Schechtman disjointification inequalities, the Kruglov property, Kolmogorov width estimates — in a straightforward way. The duality argument in Theorem 1 (constructing a conjugate system via conditional expectations, then applying the ℓ₂-estimate on X') is clean and correct. Theorem 2's explicit construction of badly approximable independent functions is elementary but effective. Theorem 5's disjointification step checks out. I went through the potential weak points the reader flagged and the stress-test examined, and I agree: no load-bearing flaw. The Kruglov property of X' is a real hypothesis, not automatic, and the authors are honest about this — Remark 3.8 shows (IR) can hold without it (ExpL_p for 1<p≤2), and Problem 3.14 flags the open case L_{2,q} for q<2 where the method breaks down because (L_{2,q})' ⊄ L₂. These are genuine limitations of the technique, not gaps in the logic. The self-citations to [29, 30, 31] are appropriate — they provide the (IR) definition and the L_p criterion being extended. Minor things: the paper could state more explicitly which Orlicz spaces are left unresolved (Remark 3.17 gives an example falling outside both Corollaries 3.7 and 3.16, and Problem 3.18 asks for a full characterization). The writing is functional but sometimes terse; a few proofs could use more signposting. This is a solid contribution for specialists in Banach space geometry and approximation theory. It deserves a serious referee.","headline":"Solid paper extending approximation rigidity of independent functions beyond L_p; deserves a serious referee","tokens_in":23542,"tokens_out":620,"would_cite":true,"duration_ms":31514,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A46","46E30","46B09"],"pacs":[],"model":"glm-5.2","headline":"Independent functions resist low-dimensional approximation in spaces near L_2","keywords":["Kolmogorov width","independent functions","symmetric function space","Kruglov property","Lorentz space","Orlicz space","lower 2-estimate","rigidity"],"falsifier":"If one could exhibit a symmetric space X satisfying a lower 2-estimate and X⊃L_2 but whose associate X' lacks the Kruglov property, and show that (IR) still holds for X, then the Kruglov condition would be revealed as sufficient but not necessary, narrowing the true criterion.","tokens_in":22514,"feed_emoji":"","tokens_out":1607,"duration_ms":93916,"temperature":0.7,"pith_summary":"The paper characterizes which symmetric function spaces X have the property (IR): that any N independent mean-zero functions of norm at least 1 cannot be approximated to within a constant error by any subspace of dimension proportional to but smaller than N. The authors prove that (IR) holds when X satisfies a lower 2-estimate, contains L_2, and its associate space X' has the Kruglov property (a condition ensuring sums of independent functions are controlled by their disjoint rearrangements). Conversely, (IR) fails when the fundamental function of X grows faster than t^{1/2} near zero, or when a copy of ℓ_q for q>2 can be found in the lattice structure of X. Applied to Lorentz spaces L_{p,q}, this yields a complete classification: (IR) holds exactly when 1<p<2 and 1≤q≤2, and fails when max{p,q}>2. The boundary case L_{2,q} for q<2 remains open.","feed_headline":"","feed_subtitle":"","key_machinery":"The positive direction constructs a biorthogonal system {g_k} in X' conjugate to the independent functions {f_k} in X, then applies the ℓ_2-estimate (guaranteed by the Kruglov property and lower 2-estimate) to invoke a lemma yielding averaged width bounds of order (1-n/N)^{1/2}. The negative direction constructs explicit sparse independent functions (taking values 0 or ±M with small probability) and uses width estimates for the Euclidean ball in ℓ_∞^N to show they can be approximated by O(N^γ)-dimensional subspaces.","core_discovery":"The central mechanism is the duality between a space X and its associate space X': the (IR) property for X reduces to whether sums of independent functions in X' satisfy an ℓ_2-type norm inequality (inequality (16)), which in turn depends on X' having the Kruglov property and X satisfying a lower 2-estimate. The Kruglov property — the requirement that a Poisson sum of independent copies of any function in X' remains in X' — is the load-bearing structural condition that allows comparing norms of independent sums with disjoint sums, and from there, applying classical Kolmogorov width estimates for Euclidean balls and octahedra. The failure results use a different mechanism: when the space is '","pith_inferences":["The open problem about L_{2,q} for q<2 suggests that the condition X⊃L_2 in Theorem 1 may be an artifact of the proof technique rather than a genuine threshold; the true dividing line might be whether the space is '2-dominated' in a weaker sense.","The connection between the Kruglov property and (IR) hints that rigidity of independent functions is fundamentally a statement about how a space handles probabilistic summation — spaces close enough to L_2 to control random sums will be rigid, while spaces that allow heavier-tailed behavior will not.","The phase transition at the L_2 boundary for Lorentz spaces mirrors the Hilbert-space optimality in classical approximation theory, suggesting that the (IR) property may be characterizable purely through the Boyd index interval containing 1/2."],"forward_implications":["For Lorentz spaces L_{p,q}, the (IR) property is now fully classified except at the boundary p=2, q<2, giving a clean phase diagram for rigidity of independent functions.","The duality mechanism — reducing (IR) for X to an ℓ_2-estimate in X' — suggests that any symmetric space whose associate satisfies a von Bahr–Esseen type inequality will inherit rigidity, potentially extending results to non-Orlicz, non-Lorentz settings.","The construction of a space X with both X and X' failing (IR) (Corollary 3.10) shows that rigidity is not self-dual, raising the question of whether L_2 is the unique self-dual rigid space.","The partial rigidity results (Proposition 3.19, Corollary 3.21) show that even when full (IR) fails, control on the support sizes or distributional properties of the independent functions can restore lower width bounds."],"fun_headline_variants":["When independent functions resist linear approximation in symmetric spaces","Kolmogorov width lower bounds for independent function sets in Lorentz and Orlicz spaces","The (IR) property: rigidity of independent functions beyond the Lp scale","Duality, the Kruglov property, and rigidity of independent functions","Which symmetric spaces make independent functions hard to approximate"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The positive results depend on the Kruglov property of the associate space X', which ensures that sums of independent functions in X' are norm-controlled by their disjoint rearrangements. This property is not automatic — for instance, exponential Orlicz spaces Exp L_p fail it for p>1 — and without it, the duality argument that produces the width lower bounds collapses entirely.","fun_headline_variants_meta":{"raw":{"variants":["When independent functions resist linear approximation in symmetric spaces","Kolmogorov width lower bounds for independent function sets in Lorentz and Orlicz spaces","The (IR) property: rigidity of independent functions beyond the Lp scale","Duality, the Kruglov property, and rigidity of independent functions","Which symmetric spaces make independent functions hard to approximate","Independent sums and the ℓ₂ inequality behind rigidity in symmetric spaces"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":674,"prompt_tokens":520,"completion_tokens":154,"prompt_tokens_details":null},"tokens_in":520,"tokens_out":154,"duration_ms":9326,"temperature":1.0,"reasoning_tokens":49,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T02:42:54.584481+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one could exhibit a symmetric space X satisfying a lower 2-estimate and X⊃L_2 but whose associate X' lacks the Kruglov property, and show that (IR) still holds for X, then the Kruglov condition would be revealed as sufficient but not necessary, narrowing the true criterion.","supporting_citations":[],"review_version":1}