{"id":"eb6cff89-6c68-4924-88d4-5c5969ac1d28","arxiv_id":"2506.04730","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted translations on L^p(G) are J-class exactly when inverse weight products decay to zero on large subsets, a condition strictly weaker than hypercyclicity.","lead":"These authors characterize when weighted translation operators on L^p spaces over locally compact groups are J-class, a dynamical property weaker than hypercyclicity. They show J-class requires only that inverse weight products vanish on large sets, yielding explicit examples that are J-class but not hypercyclic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3 needs an explicit shared-subsequence quantifier: as written, condition (i) can pick a different (n_k) for each compact Δ, while the proof requires one (n_k) common to condition (i) and condition (ii).","rationale":"I read the paper in good faith. The main theorems 2.1, 2.4, and 2.6 have proofs that are essentially correct up to minor constants and typos; Theorem 2.3's construction is valid once the quantifier over (n_k) is made common. The reader's identified weakest assumption about 'passes through compact subsets' is a real limitation but is explicitly assumed and is standard for second countable groups. The more immediately load-bearing issue is the ambiguity in Theorem 2.3, because it affects the key sufficiency result and the paper's claimed boundary between J-class and hypercyclicity. This does not change the verdict: the paper needs a clarification/amendment, consistent with the reader's CONDITIONAL recommendation.","tokens_in":10364,"tokens_out":42799,"duration_ms":429040,"concrete_test":"Reformulate Theorem 2.3 with the explicit shared-subsequence quantifier above and check that the existing proof goes through line by line. Then re-examine the original wording: if the subsequence in (i) may depend on Δ, try to produce a weight on a finite cyclic group where the even subsequence satisfies (i) for some Δ and the odd subsequence satisfies (ii) but no common subsequence works; if such a weight exists, determine whether T_{a,omega} is J-class. If no common-subsequence counterexample can arise from the group/weight structure, the concern reduces to a presentation fix.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.3 states condition (i) as 'for each compact subset Δ ... there is a sequence (E_k) ... Moreover, for some subsequence (n_k), lim ess sup_{E_k} tilde omega_{n_k}=0' and condition (ii) as 'for some compact K ... lim ess sup_K omega_{n_k}=0'. The variable (n_k) in (ii) is not explicitly quantified as the same subsequence appearing in (i). Because (i) is prefixed by 'for each Δ', the natural reading is that the subsequence may depend on Δ. In the proof, for fixed f in C_c(G) with Δ=σ(f), the authors use the subsequence supplied by (i) and then invoke (ii) along that same subsequence. If (i) is per-Δ, this step is unjustified: there may be no single subsequence along which both (i) for Δ=σ(f) and (ii) hold. The theorem should be restated with an initial 'there exists a strictly increasing sequence (n_k)' such that for every compact Δ of positive measure there exist E_k with the stated properties and, for some fixed compact K, ess sup_K omega_{n_k}->0. Under that reading the proof is coherent. This is a logical gap in the statement of the main sufficiency theorem, not a computational error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a locally compact group G and a continuous weight ω, the weighted translation operator T_{a,ω} on L^p(G), and asks when it is a J-class operator, i.e. when some extended limit set J_T(x) equals the whole space. Theorem 2.1 gives a necessary condition: if T_{a,ω} is J-class, then for every compact Δ of positive measure there are sets E_k with λ(E_k)→λ(Δ) and a subsequence n_k along which ess sup_{E_k} tildeω_{n_k}→0. Theorem 2.3 states a sufficient condition involving, in addition, a fixed compact K on which ω_{n_k}→0 in essential supremum, and identifies χ_K as a J-vector. Theorem 2.6 characterizes J_{T_{a,ω}}(0)=L^p(G) by the first condition alone when a passes through compact subsets. Theorem 2.4 addresses torsion a and claims an equivalence between J_{T_{a,ω}}(0)=L^p(G) and a local weighted-product condition. Three examples on R and R_+ are claimed to satisfy the conditions while failing hypercyclicity, supporting the paper's headline that J-class behavior is strictly weaker than hypercyclicity.","tokens_in":10639,"tokens_out":20439,"duration_ms":189397,"significance":"The topic is a natural continuation of the Costakis--Manoussos theory of J-class operators, and the paper gives a concrete framework for comparing J-class behavior with hypercyclicity for weighted translations on locally compact groups. The main approximation mechanism, based on supports separated by the 'passes through compact subsets' property, is promising, and the examples are explicit and checkable. The paper also claims J-class behavior on nonseparable L^p spaces, which is of independent interest. However, several quantifier and proof issues in the main statements need to be repaired before the results can be accepted as proved.","major_comments":[{"comment":"In Theorem 2.3, condition (i) is quantified per compact subset Δ: for each Δ there is a subsequence (n_k), while condition (ii) refers to 'some subsequence (n_k)' without specifying that it is the same subsequence. This matters in the proof, where for a fixed f with Δ=σ(f) the authors use the subsequence supplied by (i) and then apply (ii) along that same subsequence (see the line 'our hypotheses imply ... ess sup ω_{n_k} < ... on K'). With the stated quantifiers there may be no single subsequence along which both (i) and (ii) hold. The theorem should be restated with an initial quantifier: 'there exists a strictly increasing sequence (n_k) such that for every compact Δ ... and for some fixed compact K ...'. This is load-bearing for the sufficiency claim.","section":"Theorem 2.3"},{"comment":"The condition λ(E_k)→λ(Δ) should be replaced by 'E_k⊆Δ and λ(Δ\\E_k)→0'. As written, λ(E_k)→λ(Δ) does not imply λ(σ(f)\\E_k)→0 when E_k is not contained in σ(f), yet the proofs use exactly that implication: in Theorem 2.3 the integral ∫|fχ_{E_k}−f|^p is estimated by λ(σ(f)−E_k)‖f‖_∞^p, and the same use appears in Theorem 2.6. The statements should make the inclusion explicit.","section":"Theorems 2.3 and 2.6"},{"comment":"In the proof of (i)⇒(ii), the inequalities λ(B_{η,k})<η^p and λ(C_{η,k})<η^p are not justified by the hypotheses ‖g_k‖_p<ε and ‖T^{n_k}g_k−χ_Δ‖_p<ε. These hypotheses only give λ(B)<(ε/η)^p and λ(C)<(ε/η)^p. One can repair the argument by choosing the norm bounds much smaller than η^2 before fixing η, but as written the proof is invalid.","section":"Theorem 2.6, proof of (i)⇒(ii)"},{"comment":"The proof of (i)⇒(ii) begins with 'Without loss of generality, assume λ(F)>3δ^p and δ/(1−δ)<ε'. These are not harmless normalizations for arbitrary compact F and arbitrary ε,δ: one cannot enlarge F or shrink δ. In addition the final estimate gives λ(F\\E)<2δ^p, which is not the stated λ(F\\E)<δ unless 2δ^p≤δ, an extra condition that is not assumed. The converse direction (ii)⇒(i) produces for each neighborhood pair some n, but the neighborhood definition of J_T(0) requires n exceeding any prescribed N, and it is not shown that the n supplied by (ii) can be taken arbitrarily large. The equivalence is therefore not proved as stated.","section":"Theorem 2.4"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors and broken line breaks ('lo cally', 'p aper', 'defined') and should be proofread carefully.","section":"Throughout"},{"comment":"In the proof of Theorem 2.1, the estimate λ(Δ\\E)<3η^p is established for a fixed η; to obtain a sequence (E_k) with λ(E_k)→λ(Δ), the authors should explicitly let η→0 and diagonalize over k. This is routine but should be stated.","section":"Theorem 2.1"},{"comment":"In Example 2.11, the statement 'for each x∈[0,1/4] we have 1/4<ω(x)<11/16' is not correct at the endpoint x=1/4, where ω(x)=11/16. This does not affect the argument.","section":"Example 2.11"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is worth pursuing. The main issues are the quantifier gap in Theorem 2.3 and the incomplete proof of Theorem 2.4; both are repairable but affect the main claims as stated. I see no citation or novelty concerns beyond the normal need to tighten the exposition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a genuine extension of Costakis–Manoussos J-class results from weighted shifts to weighted translations on Lp(G). The necessary conditions (Theorem 2.1, and the (i)⇒(ii) half of Theorem 2.6) are argued in detail and look correct, and the examples do show J-class weighted translations that are not hypercyclic, including on nonseparable Lp(G). That part is worth having.\n\nThe soft spots are in the sufficient directions. Theorem 2.3 has a quantifier problem: condition (i) lets the subsequence (n_k) depend on the compact Δ; condition (ii) uses the same symbol without a shared quantifier. The proof needs one subsequence that works simultaneously for Δ = σ(f) and for the fixed compact K from (ii). As written, the sufficiency claim is not justified. The fix is to state upfront \"there exists a strictly increasing sequence (n_k)\" and then require (i) for every compact Δ along that sequence.\n\nThere is a second, more subtle statement issue. Conditions (i) in Theorems 2.3 and 2.6 say λ(E_k) → λ(Δ), but the proofs require the error λ(Δ \\ E_k) to be small. That is not implied by measure convergence unless the sets E_k are chosen inside Δ. The statements should explicitly require E_k ⊆ Δ (or λ(Δ \\ E_k) → 0). Without that, the condition is too weak and the estimates in the sufficiency proofs do not follow.\n\nTheorem 2.4, the torsion case, is the roughest. The (ii)⇒(i) proof produces, for each δ, some n and E, but the J-class neighborhood condition needs arbitrarily large n; that step is missing. The assertion \"ess sup_E ω_γ > 1\" also appears without justification. The torsion result may be true, but the proof as written needs real repair.\n\nMinor point: the Introduction says torsion elements \"cannot be J-class\" while the abstract and Section 2 prove the opposite; that should be corrected. No circularity problems; the main results are derived from cited external results, and the self-citations are not load-bearing.\n\nNet: the paper is a serious contribution to a small subfield, and the central claims are plausible, but the main sufficiency theorems are not yet correctly stated. A good referee could get the paper into shape with a revision. I would send it to review rather than desk-reject it.","headline":"A meaningful extension of J-class theory to weighted translations on locally compact groups, with solid necessary conditions but sufficiency statements that need quantifier and subset fixes before the main theorems are airtight.","tokens_in":11192,"tokens_out":16451,"would_cite":false,"duration_ms":142505,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A16","47B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"Weighted translations on locally compact groups are J-class exactly when one backward weight product vanishes on almost all of every compact set, and this is strictly weaker than hypercyclicity.","keywords":["J-class operator","extended limit set","weighted translation","locally compact group","hypercyclic operator","L^p space","Haar measure","linear dynamics"],"falsifier":"A single concrete counterexample would decide the boundary claim: exhibit a weighted translation $T_{a,\\omega}$ on some $L^p(G)$ that is $J$-class but for which there is a compact set $\\Delta$ of positive measure with no sequence $E_k\\subseteq\\Delta$ satisfying $\\lambda(E_k)\\to\\lambda(\\Delta)$ and $\\operatorname{ess\\,sup}_{E_k}\\tilde{\\omega}_{n_k}\\to 0$; Theorem 2.1 would then be false. One could also test a locally compact non-second-countable group with an aperiodic element $a$ that fails to pass through compact subsets and check whether the equivalence in Theorem 2.6 still holds there.","tokens_in":10162,"feed_emoji":"🌀","tokens_out":11539,"duration_ms":106525,"temperature":0.7,"pith_summary":"This paper asks when a weighted translation $T_{a,\\omega}$ on $L^p(G)$ of a locally compact group is $J$-class, meaning that for some vector the extended limit set—the set of all limits reachable by slightly perturbing the vector and letting time go to infinity—is the whole space. The authors show that, for translation elements that eventually move every compact set off itself, $J$-class behavior is controlled by one asymptotic condition: a backward weight product must tend to zero on larger and larger subsets of each compact set. That condition alone makes the extended limit set at the zero vector equal to $L^p(G)$, while hypercyclicity requires a second weight product to vanish as well. The paper constructs concrete weights for which $T_{a,\\omega}$ is $J$-class but not hypercyclic, and it shows that torsion elements, which never have dense orbits, can still satisfy $J_{T_{a,\\omega}}(0)=L^p(G)$.","feed_headline":"Weighted translations can be J-class without being hypercyclic","feed_subtitle":"For locally compact groups, one weight product vanishing on large sets suffices; hypercyclicity needs both.","key_machinery":"The machinery is the pair of weight products $\\tilde{\\omega}_m(x)=\\prod_{i=0}^{m-1}\\omega(xa^{-i})^{-1}$ and $\\omega_m(x)=\\prod_{i=1}^{m}\\omega(xa^i)$, which appear in the iterates $T_{a,\\omega}^m f(x)=\\omega(x)\\omega(xa^{-1})\\cdots\\omega(xa^{-(m-1)})f(xa^{-m})$. The extended limit set $J_T(x)$ collects the targets $y$ for which a sequence $x_n\\to x$ satisfies $T^{n_k}x_n\\to y$. The proofs use the change-of-variable formula for Haar measure to trade smallness of these products on large subsets of a compact set for approximation of arbitrary target functions, while the assumption that $a$ passes through compact subsets keeps the support of the target disjoint from the shifted support of the perturbation, so the two products can be controlled independently.","core_discovery":"The core claim is a sharp boundary between $J$-class and hypercyclicity for weighted translations on locally compact groups. Theorem 2.6 states that if $a$ passes through compact subsets, then $J_{T_{a,\\omega}}(0)=L^p(G)$ holds if and only if for every compact set $\\Delta$ of positive measure there are Borel sets $E_k\\subseteq \\Delta$ with $\\lambda(E_k)\\to\\lambda(\\Delta)$ and, along a subsequence $n_k$, $\\operatorname{ess\\,sup}_{x\\in E_k}\\tilde{\\omega}_{n_k}(x)\\to 0$. For a nonzero $J$-vector, the same condition is necessary (Theorem 2.1) and becomes sufficient in Theorem 2.3 when joined with a second condition, $\\operatorname{ess\\,sup}_{x\\in K}\\omega_{n_k}(x)\\to 0$ on some compact set $K$ of positive measure. Since the known hypercyclicity criterion for these operators requires both weight products to vanish on large sets, the paper's examples show the $J$-class condition is genuinely weaker and can even occur on non-separable $L^p(G)$, where hypercyclicity is impossible.","pith_inferences":["The asymmetry identified here suggests a testable recipe for other group actions: if an operator admits local perturbations supported away from the target, the zero-vector case should be characterized by one-sided decay of the cocycle, and two-sided decay should be required only for a dense orbit.","Because Theorem 2.1's necessary condition only involves $\\tilde{\\omega}_n$, a natural next question is whether condition (i) alone is already sufficient for a nonzero $J$-vector; the paper's sufficient condition (ii) may be an artifact of the particular proof using $\\chi_K$.","The torsion result points toward constructing J-class operators on compact groups by choosing weights whose period-$\\gamma$ product drops below 1 on large pieces of every compact set; the actual orbit remains bounded, but the extended limit set can still be everything."],"forward_implications":["If the conditions of Theorem 2.3 hold, then the indicator function of any suitable compact set $K$ is a $J$-vector, so compactly supported functions can serve as starting points for the extended limit set.","For elements passing through compact subsets, $J_{T_{a,\\omega}}(0)=L^p(G)$ is equivalent to the single backward-product condition, so the zero vector can be a $J$-vector in non-separable spaces where no dense orbit exists.","Hypercyclicity and $J$-class coincide only when both $\\tilde{\\omega}_n$ and $\\omega_n$ vanish on large sets; the examples on $\\mathbb{R}$ and $\\mathbb{R}^+$ show the gap is inhabited.","For a torsion element of order $\\gamma$, $J_{T_{a,\\omega}}(0)$ fills the whole space exactly when $\\omega_n^{-1}$ becomes arbitrarily small on almost all of every compact set, even though orbits are never dense.","Interior points of $J_{T_{a,\\omega}}(0)$ force the whole extended limit set to be $L^p(G)$, giving a binary alternative: either $J_{T_{a,\\omega}}(0)$ has empty interior or it is everything."],"supporting_citations":[{"why":"supplies the hypercyclicity criterion for weighted translations on groups and the lemma on aperiodic elements passing through compact subsets; the paper's non-hypercyclicity examples quote its theorem","marker":"[10]"},{"why":"defines the extended limit set and J-class operators and gives the power-bounded identity $J_T(x)=L_T(x)$ used in the torsion case","marker":"[12]"},{"why":"introduces J-class weighted shifts and the weight-sequence viewpoint that the main theorems adapt to group translations","marker":"[13]"},{"why":"establishes that J-class operators exist on general Banach spaces, motivating the construction of concrete J-class weighted translations","marker":"[16]"}],"fun_headline_variants":["J-class operators: a weaker condition than hypercyclicity for translations","Weighted translations: J-class without hypercyclicity, new boundary","Sharp boundary: J-class vs hypercyclic for weighted translations","J-class but not hypercyclic: weighted translations on groups","When J-class holds but hypercyclicity fails: translation example"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume the translation element moves every compact set completely away from itself after finitely many steps; if this support-separation property fails, the proof's key disjointness step collapses and no characterization is given.","fun_headline_variants_meta":{"raw":{"variants":["J-class operators: a weaker condition than hypercyclicity for translations","Weighted translations: J-class without hypercyclicity, new boundary","Sharp boundary: J-class vs hypercyclic for weighted translations","J-class but not hypercyclic: weighted translations on groups","When J-class holds but hypercyclicity fails: translation example"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1441,"prompt_tokens":991,"completion_tokens":450,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":364}},"tokens_in":607,"tokens_out":450,"duration_ms":4603,"temperature":1.0,"reasoning_tokens":364,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:37:57.917804+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A single concrete counterexample would decide the boundary claim: exhibit a weighted translation $T_{a,\\omega}$ on some $L^p(G)$ that is $J$-class but for which there is a compact set $\\Delta$ of positive measure with no sequence $E_k\\subseteq\\Delta$ satisfying $\\lambda(E_k)\\to\\lambda(\\Delta)$ and $\\operatorname{ess\\,sup}_{E_k}\\tilde{\\omega}_{n_k}\\to 0$; Theorem 2.1 would then be false. One could also test a locally compact non-second-countable group with an aperiodic element $a$ that fails to pass through compact subsets and check whether the equivalence in Theorem 2.6 still holds there.","supporting_citations":[{"cited_title":"Chen and C.-H","cited_arxiv_id":null,"evidence_quote":"supplies the hypercyclicity criterion for weighted translations on groups and the lemma on aperiodic elements passing through compact subsets; the paper's non-hypercyclicity examples quote its theorem"},{"cited_title":"Costakis, A","cited_arxiv_id":null,"evidence_quote":"defines the extended limit set and J-class operators and gives the power-bounded identity $J_T(x)=L_T(x)$ used in the torsion case"},{"cited_title":"Costakis, A","cited_arxiv_id":null,"evidence_quote":"introduces J-class weighted shifts and the weight-sequence viewpoint that the main theorems adapt to group translations"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that J-class operators exist on general Banach spaces, motivating the construction of concrete J-class weighted translations"}],"review_version":1}