REVIEW 1 major objections 4 minor 23 references
On maldistributed sequences and meager ideals
T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read An ideal is meager iff almost all sequences hit every open set.
desk verdict Answers a posed open problem with a clean equivalence, but the proof of Proposition 2.1 needs a completeness argument to be fully rigorous. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the set $\Sigma_\nu(X)=\{x\in X^\omega:\Gamma_x(\mathcal{I})=X\}$ of $\nu$-maldistributed sequences, where $\nu=1_{\mathcal{I}^+}$ is the diffuse submeasure that is $1$ on sets outside the ideal and $0$ inside. The machinery that carries the argument has three parts: Talagrand's characterization of meager ideals by the existence of long intervals meeting every $\mathcal{I}^+$-set, Laflamme's filter game in which Player II wins exactly when the ideal is meager, and the Banach–Mazur game characterization of comeager sets. Proposition 2.1 is the hinge: it uses Laflamme's game to show that if even one fiber $S_\eta=\{x:\eta\in\Gamma_x(\mathcal{I})\}$ is comeager in a Baire space $X^\omega$, then the ideal must be meager, which gives the hard direction (iii)$\Rightarrow$(ii).
What would settle it
A concrete way to refute Theorem 1.2 would be to exhibit an ideal $\mathcal{I}$ that is not meager but whose characteristic submeasure still satisfies condition (3), or, conversely, a meager ideal $\mathcal{I}$ and a Polish space $X$ with $|X|\ge 2$ for which $\Sigma_{1_{\mathcal{I}^+}}(X)$ is not comeager. The theorem predicts that neither example can exist, so either construction settles the claim.
Extended reading notes
Core claim
Theorem 1.2 is the paper's central claim: for a Polish space $X$ with at least two points and an ideal $\mathcal{I}$ on $\omega$, the following are equivalent: the characteristic submeasure $\nu=1_{\mathcal{I}^+}$ satisfies Mišík–Tóth condition (1); the ideal $\mathcal{I}$ is meager; and the set $\Sigma_\nu(X)$ of $\nu$-maldistributed sequences is comeager. A sequence is $\nu$-maldistributed precisely when every element of $X$ is an $\mathcal{I}$-cluster point of the sequence, so the theorem says that topologically small ideals are exactly the ideals for which generic sequences accumulate everywhere. The proof passes through a reformulation of condition (1) as the existence of an interval-window function $g$ such that every set in the dual filter meets $[n,n+g(n)]$ eventually, and it uses Talagrand's interval characterization of meager ideals together with Laflamme's filter game. The paper closes with Proposition 1.3, which shows that for every lower semicontinuous submeasure $\varphi$ with $\|\omega\|_\varphi=1$, the norm $\|\cdot\|_\varphi$ satisfies condition (1), so $\Sigma_{\|\cdot\|_\varphi}(X)$ is comeager for every separable metric space $X$, and with an example showing that $\Sigma_\nu(X)$ can be neither meager nor comeager.
Load-bearing premise
The whole equivalence rests on two external theorems used as black boxes—Talagrand's interval description of topologically small ideals and Laflamme's game characterization of the same property—and the argument would not go through if either statement were missing or misstated.
Editorial extensions
If this is right
- Mišík and Tóth's open problem is settled in the negative for the submeasures $1_{\mathcal{I}^+}$: their sufficient condition (1) is also necessary for $\Sigma_\nu(X)$ to be comeager, whenever $X$ is Polish with at least two points.
- Meagerness of an ideal can be detected by generic sequence behavior alone: an ideal is meager if and only if almost every $X$-valued sequence has every point of $X$ as an $\mathcal{I}$-cluster point.
- The implication 'meager ideal implies comeager $\Sigma_\nu(X)$' holds for every separable metric space $X$, not only Polish spaces.
- For every lower semicontinuous submeasure $\varphi$ with $\|\omega\|_\varphi=1$, the induced norm $\|\cdot\|_\varphi$ satisfies condition (1), so maldistributed sequences are comeager; this generalizes the known density-ideal and norm-related results.
- No simple dichotomy holds for the Baire category of $\Sigma_\nu(X)$: the paper's example splits $\omega$ into two halves generated by two maximal ideals and produces a set that is neither meager nor comeager.
Reading between the lines
- A natural extension the paper leaves open is a characterization for arbitrary diffuse submeasures $\nu$ rather than only those of the form $1_{\mathcal{I}^+}$; Proposition 1.3 supplies a sufficient condition, but nothing in the paper shows it is necessary outside the characteristic-submeasure case.
- Example 1.4 suggests that the Baire category of $\Sigma_\nu(X)$ is controlled by more than the meagerness of $\mathcal{I}$; a useful next step would be to classify which ideals make $\Sigma_\nu(X)$ meager, since comeagerness is now characterized but the intermediate behavior is not.
- Because the proof of (iii)$\Rightarrow$(ii) needs only a single comeager fiber $S_\eta$, the theorem offers a practical test for meagerness: to prove an ideal is meager, it is enough to find one point whose fiber of sequences accumulating there is comeager.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies ν-maldistributed sequences, where ν is the {0,1}-valued submeasure associated to an ideal I on ω. Theorem 1.2 proves that for a Polish space X with |X|≥2, the following are equivalent: (i) the Mišík–Tóth condition (1) holds for ν=1_{I+}; (ii) I is meager; (iii) the set Σ_ν(X) of ν-maldistributed sequences is comeager. This gives a negative answer to Mišík–Tóth's open problem for this family of submeasures. Proposition 1.3 shows that for any lower semicontinuous submeasure φ with ∥ω∥_φ=1, the submeasure ∥·∥_φ satisfies condition (1), hence the corresponding maldistributed sequences form a comeager set. An example (Example 1.4) shows that Σ may be neither meager nor comeager for a product of two maximal ideals. The proofs use Talagrand's interval characterization, Laflamme's filter game, and Banach–Mazur games.
Significance. The main result gives a clean three-way equivalence connecting a combinatorial property of ideals (meagerness) with a topological genericity property of sequences, thereby answering an open problem of Mišík and Tóth for an important class of submeasures. The proof strategy is elegant and transparent, combining standard tools in a natural way. Proposition 1.3 provides a broad and useful class of examples. The paper is generally well written and the arguments are detailed, though one significant gap appears in Proposition 2.1 as discussed below. The contribution is likely to be of interest to researchers working on ideal convergence, descriptive set theory, and topological dynamics.
major comments (1)
- [Proposition 2.1] In the proof of Proposition 2.1, the step 'We obtain by construction that there exists a sequence x = (x_n) ∈ X^ω such that x ∈ ∩_k B_k' is asserted without justification. In an arbitrary Baire space, a decreasing sequence of nonempty open sets can have empty intersection (e.g., (0,1/k) in R), and the assumption that X^ω is Baire does not rule this out. Since this x is used to identify {n : x_n ∈ U} with ∪_k F_k and thereby to conclude that Player II's strategy is winning, the proof of Proposition 2.1 is incomplete as written. Because Theorem 1.2(iii)=>(ii) invokes Proposition 2.1, the central equivalence is affected. In the application X is Polish, so X^ω is completely metrizable and the gap is repairable by choosing B_k with closure contained in G_k∩A_k and with diameter tending to 0, then applying completeness to obtain x. However, Proposition 2.1 is stated for arbitrary Hausdorff X with X^ω Baire, a hypothesis under which this repair is unavailable. The proposition should either be restricted to completely metrizable (or Polish) spaces or its proof must be completed for the Baire case.
minor comments (4)
- [Example 1.4] The final conclusion that Σ is neither meager nor comeager does not follow solely from the facts that Σ = S_{0,1} ∪ S_{1,0} and S_{0,1}, S_{1,0} are homeomorphic and disjoint. One also needs the observation that the homeomorphism which flips the odd coordinates maps Σ to its complement, so that Σ and its complement are homeomorphic; this should be stated explicitly.
- [Theorem 1.2, proof of (ii)=>(iii)] In the Banach–Mazur strategy, the phrase 'Pick j_n ∈ ω such that min I_{j_n} > κ_n' should specify that the j_n are chosen increasing (or at least with j_n → ∞). Otherwise, if κ_n is bounded, the same interval I_{j_n} could be chosen every time, and the conclusion that {n : d(x_n,η)<ε} contains infinitely many intervals I_{j_n} would not follow.
- [Throughout] There are several typos: 'Talangrand' should be 'Talagrand' in the proof of Theorem 1.2, and 'inifinitely' should be 'infinitely'.
- [Proposition 2.1] The description of W_n as 'the smallest nonempty open set such that if x∈B_k then x_n∈W_n' is imprecise; for a basic open cylinder, W_n is simply the n-th coordinate projection, and there is no canonical 'smallest' set. Also, the reduction 'without loss of generality that F_k ∩ C_{k+1} = ∅' should be justified by noting that deleting finitely many elements from F_k does not change the I^+-status of ∪_k F_k.
Circularity Check
No significant circularity: the main theorem is derived from external results (Talagrand, Laflamme, Kechris), and the paper's own prior results are either re-proved or used only in side remarks.
full rationale
The central claim (Theorem 1.2) is not circular. The equivalence (i)⇔(ii) is a direct reformulation of Talagrand's interval characterization [23, Theorem 2.1], which is cited and applied explicitly. The implication (ii)⇒(iii) is proved by an explicit Banach–Mazur strategy construction using Kechris [12, Theorem 8.33]. The implication (iii)⇒(ii) passes through Proposition 2.1, whose proof is given in full and relies on Laflamme's filter-game theorem [14, Theorem 2.12] as an external black box. The prefatory reference to [2, Theorem 3.1] before Proposition 2.1 is not load-bearing because the proposition is re-proved in the text. The side remark invoking [8, Proposition 2.11 and Theorem 6.2] is an informative aside and is not used in any proof. Proposition 1.3 is a direct epsilon-style verification of condition (1), independent of the main characterization. Even the potential gap in Proposition 2.1 concerning the existence of x ∈ ∩_k B_k would be a correctness issue, not circularity, since the conclusion is not identified with an assumption by construction.
Assumptions & free parameters
assumptions (4)
- standard math Talagrand's characterization of meager ideals: I is meager iff there exists a sequence of intervals (I_n) with max I_n < min I_{n+1} such that every S containing infinitely many I_n is I-positive.
- standard math Laflamme's filter game theorem: Player II has a winning strategy in the Laflamme game if and only if the ideal I is meager.
- standard math Banach-Mazur game characterization: a subset of a topological space is comeager if and only if Player II has a winning strategy in the Banach-Mazur game.
- domain assumption The set of I-cluster points of a sequence is closed in X.
Cite this review
Pith. "Pith review of On maldistributed sequences and meager ideals." pith.science (2026). https://pith.science/paper/4WYHXDGY
@misc{pith2026250520490,
author = {Pith},
title = {Pith review of: On maldistributed sequences and meager ideals},
year = {2026},
howpublished = {\url{https://pith.science/paper/4WYHXDGY}},
note = {Machine review of arXiv:2505.20490}
}
abstract
We show that an ideal $\mathcal{I}$ on $\omega$ is meager if and only if the set of sequences $(x_n)$ taking values in a Polish space $X$ for which all elements of $X$ are $\mathcal{I}$-cluster points of $(x_n)$ is comeager. The latter condition is also known as $\nu$-maldistribution, where $\nu: \mathcal{P}(\omega)\to \mathbb{R}$ is the $\{0,1\}$-valued submeasure defined by $\nu(A)=1$ if and only if $A\notin \mathcal{I}$. It turns out that the meagerness of $\mathcal{I}$ is also equivalent to a technical condition given by Misik and Toth in [J. Math. Anal. Appl. 541 (2025), 128667]. Lastly, we show that the analogue of the first part holds replacing $\nu$ with $\|\cdot\|_\varphi$, where $\varphi$ is a lower semicontinuous submeasure.
Reference graph
Works this paper leans on
-
[1]
A. Aveni and P. Leonetti,Most numbers are not normal, Math. Proc. Cambridge Philos. Soc. 175 (2023), no. 1, 1–11
work page 2023
-
[2]
M. Balcerzak, S. Głąb, and P. Leonetti,Another characterization of meager ideals, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM117 (2023), no. 2, Paper No. 90, 10
work page 2023
-
[3]
J. Bès, Q. Menet, A. Peris, and Y. Puig,Recurrence properties of hypercyclic operators, Math. Ann. 366 (2016), no. 1-2, 545–572
work page 2016
-
[4]
P. Borodulin-Nadzieja and B. Farkas,Analytic P-ideals and Banach spaces, J. Funct. Anal. 279 (2020), no. 8, 108702, 31
work page 2020
-
[5]
P. Borodulin-Nadzieja, B. Farkas, and G. Plebanek, Representations of ideals in Polish groups and in Banach spaces, J. Symb. Log.80 (2015), no. 4, 1268–1289
work page 2015
-
[6]
J. Bukor and J. T. Tóth,On topological properties of the set of maldistributed sequences, Acta Univ. Sapientiae Math.12 (2020), no. 2, 272–279
work page 2020
-
[7]
Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem
I. Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc.148 (2000), no. 702, xvi+177
2000
-
[8]
R. Filipów, A. Kwela, and P. Leonetti,Borel complexity of sets of ideal limit points, preprint, last updated: Apr 18, 2025 (arXiv:2411.10866)
arXiv 2025
Show all 23 references
-
[9]
Filipów and J
R. Filipów and J. Tryba,Ideal convergence versus matrix summability, Studia Math. 245 (2019), no. 2, 101–127
2019
-
[10]
Filipów and J
R. Filipów and J. Tryba,Representation of ideal convergence as a union and intersection of matrix summability methods, J. Math. Anal. Appl.484 (2020), no. 2, 123760, 21
2020
-
[11]
Just and A
W. Just and A. Krawczyk,On certain Boolean algebrasP(ω)/I, Trans. Amer. Math. Soc. 285 (1984), no. 1, 411–429
1984
-
[12]
Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol
A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995
1995
-
[13]
Kwela and P
A. Kwela and P. Leonetti,Density-like and generalized density ideals, J. Symb. Log. 87 (2022), no. 1, 228–251
2022
-
[14]
Laflamme,Filter games and combinatorial properties of strategies, Set theory (Boise, ID, 1992–1994), Contemp
C. Laflamme,Filter games and combinatorial properties of strategies, Set theory (Boise, ID, 1992–1994), Contemp. Math., vol. 192, Amer. Math. Soc., Providence, RI, 1996, pp. 51–67
1992
-
[15]
Leonetti, Strong universality, recurrence, and analytic P-ideals in dynamical systems, preprint, last updated: May 08, 2025 (arXiv:2401.01131)
P. Leonetti, Strong universality, recurrence, and analytic P-ideals in dynamical systems, preprint, last updated: May 08, 2025 (arXiv:2401.01131)
2025 arXiv
-
[16]
Leonetti,Tauberian theorems for ordinary convergence, J
P. Leonetti,Tauberian theorems for ordinary convergence, J. Math. Anal. Appl.519 (2023), no. 2, Paper No. 126798, 10
2023
-
[17]
Leonetti and F
P. Leonetti and F. Maccheroni,Characterizations of ideal cluster points, Analysis (Berlin) 39 (2019), no. 1, 19–26
2019
-
[18]
Louveau and B
A. Louveau and B. Veličković,A note on Borel equivalence relations, Proc. Amer. Math. Soc. 120 (1994), no. 1, 255–259. On maldistributed sequences and meager ideals 9
1994
-
[19]
Mišík and J
L. Mišík and J. T. Tóth,Maldistributed sequences in metric spaces, J. Math. Anal. Appl. 541 (2025), no. 2, Paper No. 128667, 7
2025
-
[20]
Olsen,Extremely non-normal numbers, Math
L. Olsen,Extremely non-normal numbers, Math. Proc. Cambridge Philos. Soc.137 (2004), no. 1, 43–53
2004
-
[21]
Solecki,Analytic ideals and their applications, Ann
S. Solecki,Analytic ideals and their applications, Ann. Pure Appl. Logic99 (1999), no. 1-3, 51–72
1999
-
[22]
Talagrand, A simple example of pathological submeasure, Math
M. Talagrand, A simple example of pathological submeasure, Math. Ann. 252 (1979/80), no. 2, 97–102
1979
-
[23]
Talagrand,Compacts de fonctions mesurables et filtres non mesurables, Studia Math.67 (1980), no
M. Talagrand,Compacts de fonctions mesurables et filtres non mesurables, Studia Math.67 (1980), no. 1, 13–43. Department of Economics, Università degli Studi dell’Insubria, via Monte Generoso 71, 21100 V arese, Italy Email address: leonetti.paolo@gmail.com
1980
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.