{"id":"347eb7fc-2ce4-4221-800c-6cc8e5c413a0","arxiv_id":"1908.05776","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Z^omega, the four cardinal invariants of Haar null and Haar meager ideals are determined in ZFC: additivity omega_1, covering min{b,cov(N)} (respectively cov(M)), uniformity max{d,non(N)} (respectively non(M)), and cofinality c.","lead":"This paper calculates the four main cardinal invariants of Haar null and Haar meager sets for the infinite product group Z^omega, all in ZFC. It resolves an open question and reveals that the cofinality of both ideals is exactly the continuum, unlike a generalized variant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Cov/non values for Z^omega rest on the imported witness-function characterization in Lemma 2.1; if the abelian case of [2, Thms 4.3/5.1/11.7] fails, Theorems 2.5/2.6 lose their foundation.","rationale":"The reader's weakest assumption pointed to the witness-function theorem from [2], and this is indeed the least secure external input for the central cov/non results. However, the concern is narrower than the reader stated: for Z^omega the non-abelian adaptation is unnecessary because both the source and quotient are abelian. The rest of the proof, especially the cofinality construction, is detailed and internally coherent; no fatal flaw or circular step was found. The left/right compact-catcher mismatch is a minor terminology issue that does not affect the argument because right compact catcher implies compact catcher, which is the property actually used to rule out Haar null/meager sets. Since the witness-function theorem is a plausible published result and the paper's own reasoning around it is sound, the concern does not warrant changing the ACCEPT verdict, though it keeps confidence at MODERATE.","tokens_in":22410,"tokens_out":32999,"duration_ms":278375,"concrete_test":"Independently prove the witness-function characterization for H=(Z/2Z)^omega: for every Borel Haar null A subset H, construct a continuous f:2^omega->H with f^{-1}(gAh) null in 2^omega for all g,h in H, and likewise meager for Haar meager A. Then re-run the pullback in Corollary 2.2 for the coordinate projection Z^omega->(Z/2Z)^omega. If the characterization is false, the lower bounds in Theorems 2.5/2.6 fail; if a proof is supplied, the concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is Lemma 2.1 together with Corollary 2.2: the upper bounds cov(HN(Z^omega))≤cov(N), cov(HM)≤cov(M) and the lower bounds non(HN)≥non(N), non(HM)≥non(M) are obtained by pulling back null/meager sets from the compact quotient H=(Z/2Z)^omega. The pullback lemma imports [2, Theorems 4.3, 5.1 and 11.7], which give a witness function f:2^omega->H for every Borel Haar null/meager set in H. For Z^omega only the abelian case is needed, so the paper's non-abelian adaptation via Michael selection is not the sensitive part; the sensitive part is the abelian witness-function characterization itself, which is cited rather than proved. If that characterization failed, both the covering-number and uniformity lower bounds in Theorems 2.5 and 2.6 would collapse. The cofinality construction in Section 2.2 is self-contained and detailed. The only internal blemish is that Definition 2.9 asks for 'left compact catcher' while Lemma 2.44 proves right compact catcher; but right compact catcher implies compact catcher, so this does not affect the argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22707,"tokens_out":12947,"duration_ms":112223,"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":[{"comment":"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":"Section 2.2, Definition 2.20"},{"comment":"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.","section":"Section 2.1, Lemma 2.1"}],"minor_comments":[{"comment":"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.","section":"Definition 2.9 and Lemma 2.44"},{"comment":"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.","section":"Proof of Theorem 2.10, first paragraph"},{"comment":"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.","section":"Lemmas 2.22, 2.23, and 2.32"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The missing inductive definition of m_ψ in Definition 2.20 looks like a cut-and-paste omission; once supplied, the construction is likely sound. The authors should also sharpen Lemma 2.1 by stating the precise imported theorem and its applicability. The mismatch between 'left compact catcher' and the right-catcher lemma is cosmetic. I would support acceptance after these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSoft 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.\n\nI 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.\n\nWho 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.","headline":"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.","tokens_in":23205,"tokens_out":2860,"would_cite":true,"duration_ms":27381,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E17","22F99","03E15","28A99"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines all four cardinal invariants of Haar null and Haar meager ideals on Z^omega, in ZFC alone.","keywords":["Haar null sets","Haar meager sets","cardinal invariants","cardinal characteristics","Cichoń Diagram","Polish groups","additivity","cofinality"],"falsifier":"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.","tokens_in":22232,"feed_emoji":"♾️","tokens_out":13577,"duration_ms":111642,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact cardinal values for Haar null and Haar meager on Z^omega","feed_subtitle":"Their covering and uniformity numbers match the real line's null and meager ideals; cofinality is c.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Banakh's computation of the four invariants for generalized Haar null sets on $\\mathbb{Z}^\\omega$ gives the comparison bounds that pin down $\\mathrm{cov}$ and $\\mathrm{non}$ for ordinary Haar null sets.","marker":"[1]"},{"why":"The witness-function theorem (Theorems 4.3, 5.1 and 11.7) is the imported result that Haar null and Haar meager sets possess continuous witnesses, which Lemma 2.1 lifts through homomorphisms.","marker":"[2]"},{"why":"Becker–Kechris Theorem 1.2.6 provides that a continuous surjective homomorphism of Polish groups is open, a step needed before Michael selection can be applied in Lemma 2.1.","marker":"[4]"},{"why":"Doležal–Vlasák's earlier computation of $\\mathrm{add}(\\mathrm{HM}(\\mathbb{Z}^\\omega))=\\omega_1$ is cited as the additivity half of Theorem 1.8.","marker":"[7]"},{"why":"Elekes–Vidnyánszky proved $\\mathrm{add}(\\mathrm{HN}(\\mathbb{Z}^\\omega))=\\omega_1$ and raised the question that the covering and uniformity theorems answer.","marker":"[10]"},{"why":"Michael's zero-dimensional selection theorem is the tool that selects the lifted witness function in the proof of Lemma 2.1.","marker":"[12]"},{"why":"Solecki's construction of many pairwise disjoint compact-catcher sets is the template for the Borel \"nice\" map $\\varphi$ used to force cofinality $\\mathfrak c$.","marker":"[13]"}],"fun_headline_variants":["ZFC cardinal invariants for Haar null and meager on Z^omega","Haar null/meager cardinals on Z^omega: add=omega1, cof=c","Exact cardinal values for Haar null/meager ideals on Z^omega","Elekes-Vidnyanszky Haar cardinal question resolved in ZFC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["ZFC cardinal invariants for Haar null and meager on Z^omega","Haar null/meager cardinals on Z^omega: add=omega1, cof=c","Exact cardinal values for Haar null/meager ideals on Z^omega","Elekes-Vidnyanszky Haar cardinal question resolved in ZFC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00153,"raw_usage":{"total_tokens":6182,"prompt_tokens":1057,"completion_tokens":5125,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":5036}},"tokens_in":673,"tokens_out":5125,"duration_ms":38085,"temperature":1.0,"reasoning_tokens":5036,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:05:45.230806+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Banakh, Cardinal characteristics of the ideal of Haar null sets, Comment","cited_arxiv_id":null,"evidence_quote":"Banakh's computation of the four invariants for generalized Haar null sets on $\\mathbb{Z}^\\omega$ gives the comparison bounds that pin down $\\mathrm{cov}$ and $\\mathrm{non}$ for ordinary Haar null sets."},{"cited_title":"Haar-$\\mathcal I$ sets: looking at small sets in Polish groups through compact glasses","cited_arxiv_id":"1803.06712","evidence_quote":"The witness-function theorem (Theorems 4.3, 5.1 and 11.7) is the imported result that Haar null and Haar meager sets possess continuous witnesses, which Lemma 2.1 lifts through homomorphisms."},{"cited_title":"Becker, A","cited_arxiv_id":null,"evidence_quote":"Becker–Kechris Theorem 1.2.6 provides that a continuous surjective homomorphism of Polish groups is open, a step needed before Michael selection can be applied in Lemma 2.1."},{"cited_title":"Doležal, V","cited_arxiv_id":null,"evidence_quote":"Doležal–Vlasák's earlier computation of $\\mathrm{add}(\\mathrm{HM}(\\mathbb{Z}^\\omega))=\\omega_1$ is cited as the additivity half of Theorem 1.8."},{"cited_title":"Elekes, Z","cited_arxiv_id":null,"evidence_quote":"Elekes–Vidnyánszky proved $\\mathrm{add}(\\mathrm{HN}(\\mathbb{Z}^\\omega))=\\omega_1$ and raised the question that the covering and uniformity theorems answer."},{"cited_title":"Michael, Selected Selection Theorems, The Amer","cited_arxiv_id":null,"evidence_quote":"Michael's zero-dimensional selection theorem is the tool that selects the lifted witness function in the proof of Lemma 2.1."},{"cited_title":"Solecki, On Haar null sets, Fund","cited_arxiv_id":null,"evidence_quote":"Solecki's construction of many pairwise disjoint compact-catcher sets is the template for the Borel \"nice\" map $\\varphi$ used to force cofinality $\\mathfrak c$."}],"review_version":1}