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A Tauberian theorem for ideal statistical convergence

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Pith's one-line read For ideals contained in the density-zero ideal, ideal statistical convergence coincides with ordinary statistical convergence.

desk verdict 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. read the letter →

arxiv 1908.04853 v1 pith:GPOMHFIF submitted 2019-08-13 math.FA math.GN

Marek Balcerzak , Paolo Leonetti This is my paper · ORCID
classification math.FAmath.GN MSC 40A3511B0554A20
keywords idealstatisticalconvergenceTauberiantheoremsubmeasuresgeneralizeddensityzeromaximalalpha-flatsequence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 3, proof of Theorem 2.4] 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.
  2. [Section 3, proof of Theorem 2.10] 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.
minor comments (5)
  1. [Proof of Theorem 2.7, final display] 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.
  2. [Proof of Corollary 2.8] 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.
  3. [After Definition 2.1] 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.
  4. [Proof of Corollary 2.9] The phrase 'the upper asymptotic density of A is most 1/2^k' should read 'at most 1/2^k'.
  5. [Proof of Theorem 2.3] 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'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Tauberian implication is proved from the hypotheses; self-citations are background only.

full rationale

The paper's main Tauberian claim (Corollary 2.8) is derived from the hypotheses I ⊆ Z via Theorem 2.7, with the representation I = Zν supplied by ν_n(A) = 1_{A∉I}, the 1-thickness of Z, and the 1-flatness of λ. None of these inputs contains the target equality. The auxiliary Theorem 2.3 defines J(I,µ) as {A : µ_n(A) →_I 0}; the existence part of the equivalence with (I,µ)-convergence is a definitional reformulation, but it does not inject any subsequent conclusion and is not used as if it were an empirical prediction; its uniqueness part is proved independently. The self-citations [19], [20], [22] are background for topology, densities, and density ideals; they are not load-bearing for Corollary 2.8, and the nonmeasurability argument in Theorem 2.10 relies on the external [17]. The representation note flagged by the reader is valid: each ν_n is a submeasure, supported on N, with ν_n({k}) = 0 because Fin ⊆ I, and limsup_n ν_n(N) = 1 > 0, so Zν = I. Thus Corollary 2.8's use of Theorem 2.7 is justified and the derivation is self-contained. Score 1 rather than 0 only because Theorem 2.3 is a by-definition existence equivalence, but it creates no circular loading of the central result.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies on standard descriptive set theory and group theory results, plus an elementary representation of ideals as Zν. No free parameters are introduced, and no new entities are postulated.

assumptions (4)
  • standard math Solecki's representation: every analytic P-ideal is Exh(ϕ) for a lower semicontinuous submeasure ϕ.
    Used in Section 2 to identify generalized density ideals as exhaustive ideals; cited from [10].
  • standard math Farah's result: every generalized density ideal is an analytic P-ideal.
    Used in Section 2 when discussing Zµ and Exh(ϕµ); cited from [11,12].
  • standard math Hernández-Hofmann-Morris: a non-closed finite-index subgroup of a compact group is nonmeasurable.
    Used in the proof of Theorem 2.10 to conclude S is nonmeasurable; cited from [17].
  • domain assumption Every proper ideal I can be represented as Zν for a smooth sequence of submeasures ν.
    Stated in Section 2 (after Definition 2.1) with a one-line construction; used in the proof of Corollary 2.8 to apply Theorem 2.7 to an arbitrary ideal I.

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Pith. "Pith review of A Tauberian theorem for ideal statistical convergence." pith.science (2026). https://pith.science/paper/GPOMHFIF

@misc{pith2026190804853,
  author       = {Pith},
  title        = {Pith review of: A Tauberian theorem for ideal statistical convergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPOMHFIF}},
  note         = {Machine review of arXiv:1908.04853}
}
abstract

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$ \textstyle \left\{n \in \mathbf{N}: \frac{1}{n}|\{k \le n: x_k \notin U\}| \ge \varepsilon\right\} \in \mathcal{I} $$ for all neighborhoods $U$ of $\ell$ and all $\varepsilon>0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}: \sum_{a \in A}1/a<\infty\}$ or the density zero ideal $\{A\subseteq \mathbf{N}: \lim_{n\to \infty} \frac{1}{n}|A\cap [1,n]|=0\}$ then $\mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $\mathcal{I}$ is maximal.

Figures

Figures reproduced from arXiv: 1908.04853 by the authors.

Figure 1
Figure 1. Graph of the sequence 1 n P i≤n xi  in the case xa3u−1 = 1. References 1. M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Generalized kinds of density and the associated ideals, Acta Math. Hungar. 147 (2015), no. 1, 97–115 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

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