{"id":"01c0a1d6-5ddd-4b94-bc17-abe59378aafd","arxiv_id":"1908.04853","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Ideal statistical convergence always equals ideal convergence for a unique derived ideal, and for ideals inside the density-zero ideal it equals statistical convergence.","lead":"The paper shows that every ideal statistical convergence notion is equivalent to ordinary ideal convergence with respect to a uniquely determined ideal, and it proves a Tauberian theorem identifying when such notions coincide with classical statistical convergence. It also shows this coincidence never happens for maximal ideals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the representation I=Zν is valid, so Corollary 2.8's use of Theorem 2.7 is justified.","rationale":"The reader's weakest assumption was that the paper does not verify the smoothness conditions for the representation I=Zν. In fact, the representation is immediate and correct: with ν_n(A)=1_{A∉I}, each ν_n is a submeasure, the support is N, the singleton values vanish because Fin⊆I, and ν_n(N)=1 because I is proper. Thus Zν=I exactly, and Corollary 2.8's appeal to Theorem 2.7 is fully justified. I also checked the main steps of the proof: Lemma 3.2 is correct, the α-flatness estimates in Claim 1 and Claim 2 of Theorem 2.7 are sound, and the thickness of Z used in Corollary 2.8 is valid. Moreover, a simple direct density argument establishes the same equivalence for any I⊆Z, which further corroborates the result. The only remaining issues are the minor typographical slips noted by the reader (interval notation in Theorem 2.7 and the notational awkwardness of 1_{1,...,k}(n) in Theorem 2.10); these do not affect correctness. Since the central claim holds and the reader's concern does not land, the verdict should remain unchanged.","tokens_in":11711,"tokens_out":25108,"duration_ms":231243,"concrete_test":"Independently verify the smooth sequence representation in Section 2: set I_n=N and ν_n(A)=1_{A∉I}. Then (s1) holds with support N, (s2) holds because Fin⊆I gives ν_n({k})=0, and (s3) holds because I is proper gives ν_n(N)=1. This confirms Zν=I exactly, and therefore the application of Theorem 2.7 in Corollary 2.8 is legitimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The reader's flagged weak point—the representation of an arbitrary proper ideal I as Zν via ν_n(A)=1_{A∉I}—is in fact valid. Checking Definition 2.1: each ν_n is a submeasure (monotone and subadditive because I is an ideal), supported on N (s1); ν_n({k})=0 because all singletons lie in Fin⊆I (s2); and ν_n(N)=1 because I is proper (s3). Hence Zν={A: limsup_n ν_n(A)=0}={A: A∈I}=I, so the hypothesis of Theorem 2.7 is satisfied. The remaining steps of Corollary 2.8 are sound: Z is 1-thick (if intervals [n,(1+c)n]⊆A for infinitely many n, then d*(A)≥c/(1+c)>0), and λ is 1-flat (|λ_n(A)−λ_{n+1}(A)|≤1/(n+1)). The central claim that I⊆Z implies equality of I-statistical and statistical convergence is therefore correct; indeed, a direct density argument gives the same result without the heavy machinery. No load-bearing mathematical gap was found.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies I-statistical convergence, a summability notion introduced by Das and Savas, and proves three main groups of results. First, for any ideal I and any smooth sequence of submeasures mu, the paper shows that (I,mu)-convergence coincides with J-convergence for a unique ideal J=J(I,mu) (Theorem 2.3), and it establishes a descriptive-complexity transfer from I to J (Theorem 2.4). Second, it proves a Tauberian-type theorem (Theorem 2.7): if Z_nu is alpha-thick and mu is alpha-flat, then (Z_nu,mu)-convergence coincides with Z_mu-convergence. As a corollary, for every ideal I contained in the density-zero ideal Z, I-statistical convergence coincides with ordinary statistical convergence (Corollary 2.8), generalizing a classical theorem of Fridy; this covers the summable ideal and the Fubini product empty-set x Fin. Third, the paper claims that for maximal ideals I, I-statistical convergence never coincides with statistical convergence (Theorem 2.10). The representation of arbitrary ideals by smooth submeasures used in the proof of Corollary 2.8 is valid: taking nu_n(A)=1_{A notin I} and I_n=N satisfies Definition 2.1.","tokens_in":11954,"tokens_out":24257,"duration_ms":249750,"significance":"If the results hold, the paper gives a clean and useful unification: I-statistical convergence is shown to be a special case of ideal convergence with an explicit ideal J(I,mu), and the Tauberian theorem in Corollary 2.8 is a genuine extension of Fridy's theorem, not a restatement of definitions. The construction of J(I,mu) and the thickness/flatness conditions are elegant, and the counterexample to the Das-Savas claim that Z-statistical convergence differs from statistical convergence is valuable. The proof of the main Tauberian theorem is self-contained and appears essentially correct. The descriptive-complexity result would also be interesting if its proof can be completed. However, as detailed in the major comments, the proofs of Theorem 2.4 (for analytic and coanalytic ideals) and Theorem 2.10 (for maximal ideals) contain gaps that are load-bearing for those advertised claims.","major_comments":[{"comment":"The displayed equality J(I,mu)=cap_m cup_k f_{m,k}^{-1}[I] only justifies the Borel case. If I is analytic, then each f_{m,k}^{-1}[I] is analytic, so cup_k is analytic, but the countable intersection cap_m of analytic sets need not be analytic. If I is coanalytic, then cup_k of coanalytic sets need not be coanalytic, and the subsequent intersection is even further from the claimed class. Thus the assertion that J is analytic (resp. coanalytic) whenever I is analytic (resp. coanalytic) is not established by the argument given. The authors should either supply an additional argument using the special structure of the submeasures or adjust the statement of Theorem 2.4.","section":"Section 3, proof of Theorem 2.4"},{"comment":"The assertion 'U is a maximal ideal on N' is false in general. For h=g circ f, the fibers are the finite sets A_{3u-2} cup A_{3u-1} cup A_{3u}, each containing at least two points. Choose B subset N containing exactly one point from each h-fiber. Then h[B]=N and h[N\\setminus B]=N, so neither B nor its complement belongs to U=h^{-1}[I], contradicting maximality of an ideal. Consequently, the claim that S has finite index is unsupported, and the appeal to [17, Proposition 1.1(c)] cannot be applied as written. The proof that J is nonmeasurable therefore has a gap that must be repaired; the theorem may be true, but the current reduction does not establish it.","section":"Section 3, proof of Theorem 2.10"}],"minor_comments":[{"comment":"The interval appearing in the inclusion for A should be N cap [n_t, n_t + c n_t^alpha], not N cap [n_t, (1+c)n_t^alpha]; the latter is not contained in the set defined by the preceding inequality unless alpha=1.","section":"Proof of Theorem 2.7, final display"},{"comment":"The sentence 'Hence Z is 1-flat' is a slight abuse of terminology: it is the sequence lambda, not the ideal Z, that is 1-flat. Please rephrase to avoid confusion.","section":"Proof of Corollary 2.8"},{"comment":"In the representation of an arbitrary ideal I as Z_mu, the support sets I_n are not specified; one should explicitly set I_n=N for all n so that condition (s1) is visibly satisfied.","section":"After Definition 2.1"},{"comment":"The phrase 'the upper asymptotic density of A is most 1/2^k' should read 'at most 1/2^k'.","section":"Proof of Corollary 2.9"},{"comment":"The notation 'lim_{n to infty} mu_n({k}) to 0' mixes a limit with an arrow; it should be written as 'lim_{n to infty} mu_n({k})=0'.","section":"Proof of Theorem 2.3"}],"recommendation":"major_revision","confidential_remarks":"The central Tauberian theorem (Theorem 2.7 and Corollary 2.8) appears sound; the reader's concern about the representation I=Z_nu is not an issue. The two gaps that block acceptance are in Theorem 2.4's descriptive-complexity claim for analytic/coanalytic ideals and in Theorem 2.10's claim that the auxiliary ideal U is maximal. Both are local to those proofs, but they are load-bearing for the respective advertised results. If the authors can repair these arguments or appropriately weaken the statements, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it embeds I-statistical convergence into a broader (I, μ)-framework, shows this always coincides with ordinary ideal convergence for a unique J, preserves descriptive complexity, and then proves a Tauberian theorem that covers the density-zero and summable ideals. The negative result for maximal ideals is a nice counterweight. The mathematics is honest and the proofs are essentially complete. The one weakness the reader flagged—the representation I = Zν via ν_n(A) = 1_{A∉I}—is actually fine. Check the definition: each ν_n is a submeasure, singletons go to 0 because Fin ⊆ I, and ν_n(N)=1 because I is proper. So the smoothness conditions hold and the application of Theorem 2.7 in Corollary 2.8 is justified. That said, the paper would be easier to read if the representation were spelled out in full instead of one sentence, especially since it is the hinge of the main corollary. The typos are minor but real: the interval notation in the proof of Theorem 2.7 should be [n_t, n_t + c n_t^α], not [n_t, (1+c)n_t^α], and the 'S not closed' justification in Theorem 2.10 could be stated more cleanly. Neither affects the conclusions. The citation pattern is fine; the self-citations are to related but distinct work and do not carry the proof weight. The paper is genuinely new: the equivalence theorem, the complexity transfer, and the Tauberian result are not in the prior literature I know. It is not a breakthrough, but it is a useful and correct contribution to summability theory and ideal convergence. I would send it to a knowledgeable referee with confidence—the referee will likely request small revisions but not find a load-bearing flaw. I would cite it if I worked on statistical convergence; the J(I, μ) construction is a clean tool worth having on record.","headline":"A solid paper that proves a genuine generalization of Fridy's Tauberian theorem; the main results hold and the one flagged gap is not actually a gap.","tokens_in":12487,"tokens_out":957,"would_cite":true,"duration_ms":11228,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["40A35","11B05","54A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For ideals contained in the density-zero ideal, ideal statistical convergence coincides with ordinary statistical convergence.","keywords":["ideal statistical convergence","statistical convergence","Tauberian theorem","submeasures","generalized density ideal","density zero ideal","maximal ideal","alpha-flat sequence"],"falsifier":"Corollary 2.8 would be refuted by a single ideal $\\mathcal{I}\\subseteq\\mathcal{Z}$ and a single real sequence that is $\\mathcal{I}$-statistically convergent to a limit but not statistically convergent to it; the natural test case is an ideal generated by sparse intervals of positive upper density, checking whether the set of times where the empirical deviation density exceeds $\\varepsilon$ can lie in $\\mathcal{I}$ while having positive density.","tokens_in":11517,"feed_emoji":"📐","tokens_out":10546,"duration_ms":95307,"temperature":0.7,"pith_summary":"The paper studies $\\mathcal{I}$-statistical convergence, a variant of statistical convergence in which the set of averaging times where the empirical density of deviations exceeds a threshold is required to belong to an ideal $\\mathcal{I}$ rather than to the finite sets. Its first result is that each such notion is not a new primitive: for any ideal $\\mathcal{I}$ and any smooth family of submeasures $\\mu$, the $(\\mathcal{I},\\mu)$-convergence is exactly ordinary convergence with respect to a uniquely determined ideal $\\mathcal{J}(\\mathcal{I},\\mu)$. The main theorem then gives a Tauberian condition for equality with ordinary statistical convergence: whenever $\\mathcal{I}\\subseteq\\mathcal{Z}$, the density-zero ideal, the two notions coincide. This extends the classical statistical Tauberian theorem, corrects an earlier published claim that $\\mathcal{Z}$-statistical convergence differs from statistical convergence, and shows that at the opposite extreme, for maximal ideals, equality never holds.","feed_headline":"Density-zero ideals make ideal statistical convergence ordinary","feed_subtitle":"Under density-zero ideals, ideal statistical convergence reduces to ordinary statistical convergence.","key_machinery":"The central object is a smooth sequence of submeasures $\\mu=(\\mu_n)$: each $\\mu_n$ is a monotone, subadditive set function with finite singleton values, supported on a nonempty set, with $\\mu_n(\\{k\\})\\to0$ for every $k$ and $\\limsup_n \\mu_n(\\mathbb{N})>0$. Such a sequence generates the ideal $\\mathcal{Z}_{\\mu}=\\{A:\\limsup_n \\mu_n(A\\cap I_n)=0\\}$, and the paper shows every ideal has this form. Two quantitative properties carry the argument: an ideal is $\\alpha$-thick if no set containing infinitely many intervals of length $cn^\\alpha$ belongs to it, and a sequence of submeasures is $\\alpha$-flat if adjacent values differ by $O(n^{-\\alpha})$ on every fixed set. The empirical measures $\\lambda_n(A)=|A\\cap[1,n]|/n$ are $1$-flat and generate the density-zero ideal, and the density-zero ideal is $1$-thick; Theorem 2.7 converts these facts into the Tauberian implication that ideal-small empirical densities are genuinely small.","core_discovery":"The paper establishes that $\\mathcal{I}$-statistical convergence is controlled by the derived ideal $\\mathcal{J}(\\mathcal{I},\\mu)=\\{A\\subseteq\\mathbb{N}: \\mu_n(A)\\to_{\\mathcal{I}}0\\}$: convergence of a sequence to a point is equivalent to membership of each off-neighborhood set in this ideal, and $\\mathcal{J}$ is unique. With lower semicontinuous submeasures, the derived ideal inherits the Borel, analytic, or coanalytic complexity of $\\mathcal{I}$. The key transfer result is that if $\\mathcal{Z}_{\\nu}$ is $\\alpha$-thick and $\\mu$ is $\\alpha$-flat, then $(\\mathcal{Z}_{\\nu},\\mu)$-convergence coincides with $\\mathcal{Z}_{\\mu}$-convergence, forcing $\\mu_n(A)\\to_{\\mathcal{Z}_{\\nu}}0$ to imply $\\mu_n(A)\\to0$ for every set $A$. Since the empirical density measures are $1$-flat and generate $\\mathcal{Z}$, and any ideal contained in $\\mathcal{Z}$ is $1$-thick, Corollary 2.8 follows: $\\mathcal{I}$-statistical convergence is statistical convergence whenever $\\mathcal{I}\\subseteq\\mathcal{Z}$. Conversely, for a maximal ideal $\\mathcal{I}$ the derived ideal is nonmeasurable while $\\mathcal{Z}$ is $\\mathrm{F}_{\\sigma\\delta}$, so equality is impossible.","pith_inferences":["The $\\alpha$-parameters suggest an immediate testable extension: lacunary statistical convergence with block lengths growing like $n^\\beta$ should satisfy the same Tauberian equivalence whenever the governing ideal is $\\alpha$-thick with $\\alpha$ matching the block growth.","The uniqueness of $\\mathcal{J}(\\mathcal{I},\\mu)$ frames ideal convergence as a change of base ideal: one may expect inclusion relations between ideals to translate, in a functorial way, into strength comparisons between the corresponding statistical convergence methods.","The nonmeasurability obstruction for maximal ideals suggests that the barrier to Tauberian restoration is descriptive complexity rather than mere size; distinguishing meager ideals from those without the Baire property might refine the classification.","Example 2.11 indicates that any comparison of 'ideal statistical convergence' across papers must specify the submeasure sequence; this dependence might motivate a canonical choice of submeasures for ideals with a natural density."],"forward_implications":["Every $\\mathcal{I}$-statistical convergence notion is ordinary convergence to a unique derived ideal $\\mathcal{J}(\\mathcal{I},\\lambda)$, so questions about these methods reduce to ideal inclusion and complexity.","For $\\mathcal{I}\\subseteq\\mathcal{Z}$, including the summable ideal and the Fubini product $\\emptyset\\times\\mathrm{Fin}$, $\\mathcal{I}$-statistical convergence coincides with statistical convergence.","For maximal ideals $\\mathcal{I}$, $\\mathcal{I}$-statistical convergence never coincides with statistical convergence.","All upper densities that dominate asymptotic density give ideals of the form $\\{A:\\mu^*(A)=0\\}$ for which $\\mathcal{I}$-statistical convergence is ordinary statistical convergence.","The method depends on the chosen submeasures, not only on the ideal they generate: two smooth sequences can generate the same ideal $\\mathcal{Z}$ while producing different convergence notions."],"supporting_citations":[{"why":"Introduces $\\mathcal{I}$-statistical convergence, the notion investigated here, and contains the remark about $\\mathcal{Z}$-statistical convergence that the paper corrects.","marker":"[6]"},{"why":"Provides the classical Tauberian condition for statistical convergence that Theorem 2.7 and Corollary 2.8 generalize.","marker":"[13]"},{"why":"Supplies the representation of analytic P-ideals as exhaustive ideals of lower semicontinuous submeasures, used for the general setting.","marker":"[10]"},{"why":"Defines generalized density ideals $\\mathcal{Z}_{\\mu}$ and their basic theory, the framework for Theorem 2.7.","marker":"[11]"},{"why":"Treats the density zero ideal in this framework, supporting the identification of $\\mathcal{Z}$ with the empirical-density ideal.","marker":"[12]"},{"why":"Gives the nonmeasurable-subgroup result used to prove the derived ideal is nonmeasurable for maximal $\\mathcal{I}$.","marker":"[17]"},{"why":"Provides the related summability-method formulations, including the lacunary case, that fit inside the $(\\mathcal{I},\\mu)$-convergence scheme.","marker":"[5]"}],"fun_headline_variants":["Density-zero ideals collapse ideal statistical convergence to ordinary","Tauberian theorem: density-zero ideals make ideal convergence statistical","Ideal statistical convergence reduces to statistical under density-zero ideals","Maximal ideals prevent the Tauberian simplification of convergence","A Tauberian bridge: density-zero ideals force statistical convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reduction rests on the assertion that every proper ideal can be represented as $\\mathcal{Z}_{\\nu}$ for a smooth sequence of submeasures; the paper states the formula $\\nu_n(A)=\\mathbf{1}_{A\\notin\\mathcal{I}}$ but does not check every smoothness condition in detail.","fun_headline_variants_meta":{"raw":{"variants":["Density-zero ideals collapse ideal statistical convergence to ordinary","Tauberian theorem: density-zero ideals make ideal convergence statistical","Ideal statistical convergence reduces to statistical under density-zero ideals","Maximal ideals prevent the Tauberian simplification of convergence","A Tauberian bridge: density-zero ideals force statistical convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000838,"raw_usage":{"total_tokens":3725,"prompt_tokens":1091,"completion_tokens":2634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":2551}},"tokens_in":707,"tokens_out":2634,"duration_ms":19745,"temperature":1.0,"reasoning_tokens":2551,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:32:21.076719+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Corollary 2.8 would be refuted by a single ideal $\\mathcal{I}\\subseteq\\mathcal{Z}$ and a single real sequence that is $\\mathcal{I}$-statistically convergent to a limit but not statistically convergent to it; the natural test case is an ideal generated by sparse intervals of positive upper density, checking whether the set of times where the empirical deviation density exceeds $\\varepsilon$ can lie in $\\mathcal{I}$ while having positive density.","supporting_citations":[{"cited_title":"Das and E","cited_arxiv_id":null,"evidence_quote":"Introduces $\\mathcal{I}$-statistical convergence, the notion investigated here, and contains the remark about $\\mathcal{Z}$-statistical convergence that the paper corrects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical Tauberian condition for statistical convergence that Theorem 2.7 and Corollary 2.8 generalize."},{"cited_title":"Farah, Analytic quotients: theory of liftings for quotients over an alytic ideals on the integers, Mem","cited_arxiv_id":null,"evidence_quote":"Supplies the representation of analytic P-ideals as exhaustive ideals of lower semicontinuous submeasures, used for the general setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines generalized density ideals $\\mathcal{Z}_{\\mu}$ and their basic theory, the framework for Theorem 2.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Treats the density zero ideal in this framework, supporting the identification of $\\mathcal{Z}$ with the empirical-density ideal."},{"cited_title":"Hernández, K","cited_arxiv_id":null,"evidence_quote":"Gives the nonmeasurable-subgroup result used to prove the derived ideal is nonmeasurable for maximal $\\mathcal{I}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the related summability-method formulations, including the lacunary case, that fit inside the $(\\mathcal{I},\\mu)$-convergence scheme."}],"review_version":1}