Cardinal invariants of Haar null and Haar meager sets

M\'ark Po\'or; M\'arton Elekes

REVIEW 2 major objections 4 minor 13 references

Cardinal invariants of Haar null and Haar meager sets

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper determines all four cardinal invariants of Haar null and Haar meager ideals on Z^omega, in ZFC alone.

desk verdict Full ZFC computation of the four cardinal invariants for Haar null and Haar meager ideals on Z^omega, with the genuinely new part being cofinality = c and the Borel 'nice group' construction behind it. read the letter →

arxiv 1908.05776 v4 pith:M2ALAZPV submitted 2019-08-15 math.LO math.GN

Márton Elekes , Márk Poór This is my paper · ORCID

classification math.LOmath.GN MSC 03E1722F9903E1528A99

keywords HaarnullsetsmeagercardinalinvariantscharacteristicsCichońDiagramPolishgroupsadditivitycofinality

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

This paper determines how many "negligible" sets are needed to cover a large one in the group $G=\mathbb{Z}^\omega$, the product of countably many copies of the integers and the simplest Polish group that is not locally compact. The two notions of negligible are Haar null sets, the non-locally-compact analogue of measure-zero sets, and Haar meager sets, the analogue of meager sets. The paper proves in ZFC alone that their four cardinal invariants—additivity, covering number, uniformity, and cofinality—take the values $\mathrm{add}=\omega_1$ for both, $\mathrm{cov}(\mathrm{HN})=\min\{\mathfrak b,\mathrm{cov}(\mathcal N)\}$, $\mathrm{cov}(\mathrm{HM})=\mathrm{cov}(\mathcal M)$, $\mathrm{non}(\mathrm{HN})=\max\{\mathfrak d,\mathrm{non}(\mathcal N)\}$, $\mathrm{non}(\mathrm{HM})=\mathrm{non}(\mathcal M)$, and $\mathrm{cof}=\mathfrak c$ for both, where $\mathcal N$ and $\mathcal M$ are the null and meager ideals of the real line and $\mathfrak b,\mathfrak d$ are the bounding and dominating numbers. These values come from the standard Cichoń characteristics, so the smallness structure of $G$ is fully calibrated without any extra set-theoretic axioms. The results answer a question of Elekes and Vidnyánszky and reveal a sharp contrast with the generalized Haar null ideal, whose cofinality can exceed $\mathfrak c$ under Martin's Axiom.

What carries the argument

The load-bearing object is the notion of a "nice" Polish group: a Borel map $\varphi:2^\omega\to \mathcal F(G)$ assigning to each binary sequence a closed set that is simultaneously Haar null and Haar meager, with the property that the union $\bigcup_{x\in P}\varphi(x)$ over any non-empty perfect $P\subset 2^\omega$ is left compact catcher—meaning some right translate of every compact set lies inside it. The paper proves that every non-locally compact Polish group with a two-sided invariant metric is nice by building $\varphi$ from a countable poset of labeled trees, metric balls $B(\delta_i)$, and specially chosen group elements, arranging that each $\varphi(c)$ is a Cantor-set-like intersection of closed balls whose two-sided translates have measure zero with respect to the coin-tossing measure on a Cantor set. Niceness is then converted into cofinality $\mathfrak c$ by a coanalytic perfect-set argument: any Borel Haar null set can contain $\varphi(x)$ for at most $\omega_1$-many $x$, so fewer than $\mathfrak c$ small sets cannot be cofinal. A second mechanism, the witness-function lifting lemma, transfers covering and uniformity numbers through continuous surjective homomorphisms using the zero-dimensional Michael selection theorem.

What would settle it

To test the covering equalities, attempt to cover $\mathbb{Z}^\omega$ by fewer than $\min\{\mathfrak b,\mathrm{cov}(\mathcal N)\}$ Borel Haar null sets, or by fewer than $\mathrm{cov}(\mathcal M)$ Borel Haar meager sets; any such cover refutes Theorems 1.6 and 1.8. To test the cofinality theorem, look for a non-locally compact Polish group with a two-sided invariant metric whose Haar null or Haar meager ideal has a cofinal family of size below $\mathfrak c$; the paper's "niceness" argument says none exists.

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Extended reading notes

Core claim

The paper establishes, in ZFC, the exact values $\mathrm{add}(\mathrm{HN}(\mathbb{Z}^\omega))=\mathrm{add}(\mathrm{HM}(\mathbb{Z}^\omega))=\omega_1$, $\mathrm{cov}(\mathrm{HN}(\mathbb{Z}^\omega))=\min\{\mathfrak b,\mathrm{cov}(\mathcal N)\}$, $\mathrm{cov}(\mathrm{HM}(\mathbb{Z}^\omega))=\mathrm{cov}(\mathcal M)$, $\mathrm{non}(\mathrm{HN}(\mathbb{Z}^\omega))=\max\{\mathfrak d,\mathrm{non}(\mathcal N)\}$, $\mathrm{non}(\mathrm{HM}(\mathbb{Z}^\omega))=\mathrm{non}(\mathcal M)$, and $\mathrm{cof}(\mathrm{HN}(\mathbb{Z}^\omega))=\mathrm{cof}(\mathrm{HM}(\mathbb{Z}^\omega))=\mathfrak c$. It further proves the additivity and cofinality equalities for every non-locally compact Polish group admitting a two-sided invariant metric, and the Haar meager covering and uniformity equalities for every Polish group admitting a continuous surjective homomorphism onto a non-discrete locally compact Polish group, which includes separable infinite-dimensional Banach spaces. This answers a question posed by Elekes and Vidnyánszky, and sharply contrasts with the generalized Haar null ideal, whose cofinality can be larger than $\mathfrak c$ under Martin's Axiom.

Load-bearing premise

The load-bearing premise is the imported witness-function theorem: every Haar null or Haar meager set must admit a continuous witness map from the Cantor space, and the paper needs this to survive the passage from abelian to arbitrary Polish groups via open mappings and Michael selection; if that adaptation fails, the covering and uniformity equalities lose their foundation.

Editorial extensions

If this is right

For $G=\mathbb{Z}^\omega$, the Haar null ideal has $\mathrm{cov}=\min\{\mathfrak b,\mathrm{cov}(\mathcal N)\}$ and $\mathrm{non}=\max\{\mathfrak d,\mathrm{non}(\mathcal N)\}$, so its covering and uniformity are controlled by the same two Cichoń characteristics that govern generalized Haar null sets.
The Haar meager ideal on $\mathbb{Z}^\omega$ has the same covering number and uniformity as the meager ideal of the real line, so set-theoretic axioms that change $\mathrm{cov}(\mathcal M)$ change the Haar meager covering number accordingly.
Cofinality of both ideals is always the continuum $\mathfrak c$; in particular neither $\mathrm{HN}(\mathbb{Z}^\omega)$ nor $\mathrm{HM}(\mathbb{Z}^\omega)$ has a cofinal family of size less than $\mathfrak c$.
For every non-locally compact Polish group admitting a two-sided invariant metric, the additivity and cofinality calculations remain valid: $\mathrm{add}(\mathrm{HN})=\mathrm{add}(\mathrm{HM})=\omega_1$ and $\mathrm{cof}(\mathrm{HN})=\mathrm{cof}(\mathrm{HM})=\mathfrak c$.
Any Polish group admitting a continuous surjective homomorphism onto a non-discrete locally compact Polish group inherits $\mathrm{cov}(\mathrm{HM})=\mathrm{cov}(\mathcal M)$ and $\mathrm{non}(\mathrm{HM})=\mathrm{non}(\mathcal M)$; examples include $\mathbb{Z}^\omega$ and separable infinite-dimensional Banach spaces.

Reading between the lines

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

One testable extension is to check whether Banakh's generalized Haar null computation extends to every separable infinite-dimensional Banach space; if it does, the same $\mathrm{cov}$ and $\mathrm{non}$ equalities proved here for $\mathbb{Z}^\omega$ would follow for those groups as well.
The contrast between $\mathrm{cof}(\mathrm{HN})=\mathfrak c$ and the generalized Haar null ideal having $\mathrm{cof}>\mathfrak c$ under Martin's Axiom suggests that the Borel witness requirement, rather than the measure alone, is what stabilizes cofinality; interpolating between the two ideals could locate exactly where the cofinality jumps.
Because the "nice" construction only needs a two-sided invariant metric and dense sequences of small balls, the same labeled-tree method may apply to other witness-generated $\sigma$-ideals, such as the Haar-I sets of reference [2], yielding ZFC computations of their cofinalities.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline is simple: this paper closes the Z^omega case for the four cardinal invariants of Haar null and Haar meager ideals, and the genuinely new content is the cofinality result (both equal c) and the 'nice group' Borel construction behind it. The additivity values were already known, and the covering/uniformity numbers are adaptations of Banakh's work on generalized Haar null sets, but the adaptation is nontrivial because the Borel-definition ideal is stricter; the paper goes through the arguments carefully.

The cofinality proof is the real meat. The construction of the Borel map phi: 2^omega -> F(G) is long and technical, with lemmas that check each property: closedness of phi(c), Haar nullness via a Cantor measure, and the perfect-set compact-catching property. I read through the main steps and they hold up. The separation estimates (Lemma 2.37, Corollary 2.38) are the core, and the proof is detailed enough to follow. This is a solid piece of work.

Soft spots, in proportion. The most load-bearing external input is the witness-function characterization from Banakh et al. [2], imported via Lemma 2.1. The cov/non values for Z^omega literally rest on that black box. The paper adapts it to non-abelian groups using Michael selection, but the abelian version is cited, not proved. If that characterization were wrong, Theorems 2.5 and 2.6 would collapse. That is a real dependency, but it is also completely normal: the paper flags it and the external results are published/arXiv. A referee should check those citations, not reject on that basis. Also minor: several small lemmas are left to the reader, the closed-Haar-null-to-Haar-meager observation gets a one-line sketch, and there is a left/right compact catcher wording mismatch that doesn't matter because right implies both-sided.

I disagree with any claim that the paper is merely derivative; the cofinality c result contrasts sharply with Banakh's generalized Haar null case where cof > c under MA, and the nice-group construction is new. The paper is honest about what is borrowed.

Who should read this? Set theorists and descriptive set theorists working on ideals on Polish groups. It deserves a serious referee; the technical core is sound as far as I can tell, and a careful referee should fill in the skipped lemmas and verify the external theorems before signing off. I would send it to peer review.

Referee Report

2 major / 4 minor

Summary. This paper determines, in ZFC, the four cardinal invariants (add, cov, non, cof) of the σ-ideals of Haar null and Haar meager sets for G=Z^ω. The main theorems state add(HN)=add(HM)=ω1, cov(HN)=min{b,cov(N)}, cov(HM)=cov(M), non(HN)=max{d,non(N)}, non(HM)=non(M), and cof(HN)=cof(HM)=c. The proofs combine a pullback lemma for continuous surjective homomorphisms (Section 2.1), an o-boundedness argument for the null ideal, and a detailed Borel construction of many compact-catching closed sets for cofinality (Section 2.2). The paper also claims extensions to groups admitting a two-sided invariant metric and to groups surjecting onto a non-discrete locally compact group.

Significance. The results, if correct, fully calibrate the smallness ideals of the simplest non-locally compact Polish group and answer a question of Elekes and Vidnyánszky. They provide a sharp contrast with Banakh's generalized Haar null case, where cofinality can be larger than the continuum under Martin's Axiom. The paper's contribution is substantial: the cofinality construction is largely self-contained and detailed, the Borelness of the map φ is proved, and the paper explicitly handles the non-abelian modification of one imported witness-function theorem. The reliance on standard descriptive set theory and on previously published results is clearly indicated, and the paper gives concrete open problems.

major comments (2)

[Section 2.2, Definition 2.20] The embedding m_ψ is announced but never actually defined; after stating the condition |m_ψ| = |n_ψ|, the text immediately moves to fixing the metric d. This m_ψ is used in Definition 2.33 for F_ψ^c and throughout the cofinality construction, so as written the construction is ill-defined. Please provide the inductive definition of m_ψ, or explicitly state that an arbitrary order-preserving injection of the countable poset T into (ω^{<ω},⊂) is fixed, and verify the injectivity property used in Lemma 2.37 and Lemma 2.45(v).
[Section 2.1, Lemma 2.1] The proof of this key pullback lemma delegates the existence of the witness function f to [2, Theorem 4.3 & Proposition 5.1] and states without proof that these results are 'completely straightforward to adapt' to non-abelian groups. Since Corollary 2.2 and hence Theorems 2.5 and 2.6 depend on this lemma, please state precisely which theorem is being imported and either prove the non-abelian adaptation or restrict Corollary 2.2 to the abelian instances actually needed for Z^ω and separable Banach spaces. The current 'essentially yields' wording leaves an unverified load-bearing step.

minor comments (4)

[Definition 2.9 and Lemma 2.44] Definition 2.9 requires ∪_{x∈P} φ(x) to be 'left compact catcher', but Lemma 2.44 proves the right-sided version (Kh ⊂ ∪_{c∈P} φ(c)). Since right compact catcher implies compact catcher, the argument is unaffected, but the terminology should be aligned for consistency.
[Proof of Theorem 2.10, first paragraph] The statement that a closed Haar null set is Haar meager is justified by a one-line sketch; please expand this into a few sentences or give a precise reference to a published proof.
[Lemmas 2.22, 2.23, and 2.32] These lemmas are stated with proofs left to the reader; given that they are used in subsequent metric estimates and tree arguments, a few lines of proof or a reference would improve the self-containedness of the paper.
[References] The paper cites [2] by arXiv version and theorem numbers 'as in Version 4'; since the published version may have different numbering, please add a note or update the reference to the final published form.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ZFC cardinal-invariant computations are assembled from external theorems (Banakh, Doležal–Vlasák, the witness-function lemma from Banakh–Głąb–Jabłońska–Swaczyna) and a self-contained cofinality construction.

full rationale

The paper's derivation chain has no step in which a 'prediction' is equivalent to its inputs by construction. The main cov and non results for Haar null sets (Theorem 2.5) combine Banakh's external Theorem 1.5 for generalized Haar null sets with Corollaries 2.2 and 2.4; the latter are obtained from the preimage lemma (Lemma 2.1) and Banakh's o-bounded results. Lemma 2.1 is imported from the external paper [2] (Banakh–Głąb–Jabłońska–Swaczyna), with a non-abelian adaptation via Michael selection; it is a dependency on prior work, not a fitted parameter or a renamed version of the target theorem. The Haar meager cov and non equalities (Theorem 2.6) likewise use the standard inclusion HM⊂M plus Corollary 2.2. The cofinality result (Theorem 2.7) is supported by a genuinely self-contained construction: Definition 2.9 introduces a 'nice' group, Theorem 2.10 constructs the Borel map phi with Haar-null/meager fibers whose perfect unions are compact catcher, and Theorem 2.11 derives cof=c from this using the perfect-set property of coanalytic sets. The self-citations ([9], [10]) are used only for prior published results or standard survey facts, not as unverified load-bearing assumptions that reproduce the paper's conclusions. The derivation is accordingly self-contained relative to its external inputs. There is a minor internal wording issue: Definition 2.9 asks for 'left compact catcher' while Lemma 2.44 establishes a right compact catcher inclusion, but the paper correctly notes that this still gives compact catcher; this is a harmless local mismatch, not circularity. The possible fragility of the imported witness-function theorem for arbitrary Polish groups is a correctness risk, not a circularity risk.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities appear; the paper is a pure proof. The axioms listed are standard descriptive set theory facts plus prior theorems on Haar null and Haar meager ideals. The most substantive imported input is the witness-function theorem from reference [2] used in Lemma 2.1.

assumptions (6)

domain assumption A Polish group is locally compact iff it carries a Haar measure; in the locally compact case Haar null sets are the measure-zero sets and Haar meager sets are the meager sets.
Invoked in the introduction and in Corollary 2.2 to transfer cov and non from locally compact groups via continuous surjective homomorphisms.
domain assumption Banakh's theorem: add(HNgen(Z^omega))=add(N), cov(HNgen(Z^omega))=min{b,cov(N)}, non(HNgen(Z^omega))=max{d,non(N)}, and under Martin's Axiom cof(HNgen(Z^omega))>c.
Used in Theorem 2.5 to get one direction of the Haar null covering and uniformity equalities.
domain assumption Banakh-Glab-Jablonska-Swaczyna witness-function theorem ([2] Theorems 4.3, 5.1, 11.7): Haar null and Haar meager sets have witness functions from 2^omega, adapted to non-abelian groups in Lemma 2.1.
This is the key imported black box for proving that preimages of Haar null and Haar meager sets under continuous surjective homomorphisms remain Haar null and Haar meager.
standard math Every coanalytic set is the union of omega_1 many Borel sets, and every uncountable Borel subset of 2^omega contains a perfect set.
Used in the proof of Theorem 2.11 to turn a coanalytic set of cardinality greater than omega_1 into a perfect set P so that the nice-map condition applies.
standard math Zero-dimensional Michael Selection Theorem: a lower semi-continuous closed-valued and nonempty-valued multifunction from a zero-dimensional space admits a continuous selection.
Used in Lemma 2.1 to lift a witness function f:2^omega to H through the open continuous homomorphism phi:G to H.
standard math In a complete metric non-locally compact group, every neighborhood of the identity is not totally bounded, so it admits no finite net; this anchors the recursive construction in Lemma 2.24.
Needed to choose delta_i, epsilon_i, and q_j^(i) with no finite cover by small-diameter sets, which drives Lemmas 2.26 and 2.28.

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Cite this review

Pith. "Pith review of Cardinal invariants of Haar null and Haar meager sets." pith.science (2026). https://pith.science/paper/M2ALAZPV

@misc{pith2026190805776,
  author       = {Pith},
  title        = {Pith review of: Cardinal invariants of Haar null and Haar meager sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2ALAZPV}},
  note         = {Machine review of arXiv:1908.05776}
}

abstract

A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $\mu$ and a Borel set $B$ containing $X$ such that $\mu(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $\sigma$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Z. Vidny\'anszky.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[2]

Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

T. Banakh, S. Gł¸ ab, E. Jabłońska, J. Swaczyna, Haar- I sets: looking at small sets in Polish groups through compact glasses, available at https://arxi v.org/abs/1803.06712

work page Pith review arXiv
[1]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

T. Banakh, S. Gł¸ ab, E. Jabłońska, J. Swaczyna, Haar- I sets: looking at small sets in Polish groups through compact glasses, available at https://arxi v.org/abs/1803.06712

work page Pith review arXiv

[2] [1]

Banakh, Cardinal characteristics of the ideal of Haar null sets, Comment

T. Banakh, Cardinal characteristics of the ideal of Haar null sets, Comment. Math. Univ. Carolin. 45 (2004), no. 1, 119–137

work page 2004

[3] [3]

Bartoszyński, H

T. Bartoszyński, H. Judah, Set theory. On the structure of the real line. A K Peters, Ltd., W ellesley, MA, 1995

work page 1995

[4] [4]

Becker, A

H. Becker, A. Kechris, The Descriptive Set Theory of Polish Group Actions , Cambridge Univ. Press, Cambridge, 1996

work page 1996

[5] [5]

J. P. R. Christensen, On sets of Haar measure zero in abeli an Polish groups, Israel J. Math. 13 (1972) 255–260

work page 1972

[6] [6]

U. B. Darji, On Haar meager sets, Topology Appl. 160 (18) (2013) 2396–2400

work page 2013

[7] [7]

Doležal, V

M. Doležal, V. Vlasák, Haar meager sets, their hulls, and relationship to compact sets, J. Math. Anal. Appl. 446 (1) (2017) 852–863

work page 2017

[8] [8]

Dougherty, Examples of non-shy sets, Fund

R. Dougherty, Examples of non-shy sets, Fund. Math. 144 (1994), 73–88

work page 1994