REVIEW 2 major objections 4 minor 27 references
On the complexity of upper frequently hypercyclic vectors
T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Upper frequently hypercyclic vector sets are always Gδσ, and can fail to be Fσδ.
desk verdict Real result with two fixable gaps in the lower-bound proof: the Gδσ upper bound is clean, the negative answer to Bonilla–Grosse-Erdmann is credible, but Theorem 3.6 needs a corrected density argument before the paper is publishable as is. 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 engine is a descriptive-set-theoretic transfer along the map from a point x to its return-time set S(U) = {n : Tnx ∈ U}. The upper-bound half uses the characterization of analytic P-ideals as Exh(φ), the family of sets whose φ-mass at infinity is zero, for a lower semicontinuous submeasure φ; together with a general theorem saying that when the forbidden family F of return-time sets belongs to the modified Borel class Π̂03, the set of points whose return sets avoid F is Gδσ. The lower-bound half is a concrete continuous reduction f : Δ → X from a closed subset Δ of the Baire space: f interleaves long zero blocks with blocks of a pointwise frequently hypercyclic vector y, rescaled so that iterates of f(x) reproduce exactly the iterates of y at selected times. This makes the set f−1[UFHC(Bw)] equal to D = Δ \ C3, a known Gδσ-but-not-Fσδ set, so UFHC itself cannot be Fσδ.
What would settle it
Read the displayed chain in Claim 4 and test whether the step writing a limsup of a product as d⋆(Si) times a liminf of the ratios is valid; if a counterexample to that inequality exists, the proof of Theorem 2.1 is incomplete as written and the theorem needs a corrected estimate, while the claim itself would be falsified if some weighted shift satisfying (2.1) on ℓp with the product topology had a UFHC set that is Fσδ.
Extended reading notes
Core claim
The paper's central claim is twofold. First, Theorem 1.2: for every continuous linear operator T on a second countable topological vector space and every analytic P-ideal I, the set HCT(I) of I-hypercyclic vectors is a Gδσ-set; in particular, the set UFHC(T) of upper frequently hypercyclic vectors is Gδσ. Second, Theorem 2.1: there exists a unilateral weighted backward shift on ℓp with p ≥ 1, with the product topology, under the summability condition ∑n (w0⋯wn)−p < ∞, for which UFHC(Bw) is not an Fσδ-set; consequently, for λ > 1 the operator λB on ℓp has UFHC(λB) not Gδ, settling the 2018 question in the negative. The proof constructs a continuous map f from a closed subset Δ of the Baire space such that f−1[UFHC(Bw)] equals a set D that is Gδσ but not Fσδ, transferring the non-complexity upward. The final results show that for countably generated ideals, norm, weak, and pointwise I-hypercyclicity coincide for weighted shifts on c0 or ℓp, and that for the ideal of density-zero sets these notions separate: a shift can be pointwise upper frequently hypercyclic and norm hypercyclic while failing to be norm upper frequently hypercyclic.
Load-bearing premise
The proof that the bad example exists depends on a cited characterization saying that a summability condition on the weight sequence makes the weighted shift frequently hypercyclic, so that a vector with all return-time sets of positive upper density exists; the manuscript also prints a density estimate in the proof that contains a false inequality, so the argument as written needs a repair to support the theorem.
Editorial extensions
If this is right
- The previously known Gδσδ upper bound for analytic P-ideals is sharpened to Gδσ, and the new example shows the bound is tight in the strong sense that the set need not be Fσδ.
- The 2018 open question of whether UFHC(T) is always Gδ is settled negatively: for λ > 1, the operator λB on ℓp with the product topology has a UFHC set that is not Gδ.
- For weighted shifts on c0 or ℓp, ideal hypercyclicity with respect to any countably generated ideal does not depend on whether the topology is norm, weak, or product.
- For the ideal of asymptotic density zero sets, pointwise upper frequent hypercyclicity is strictly weaker than norm upper frequent hypercyclicity in general: there is a shift that is pointwise upper frequently hypercyclic but not norm upper frequently hypercyclic.
- The set of upper frequently hypercyclic vectors remains either empty or comeager, so the complexity increase from Gδ to Gδσ reflects a genuine structural change rather than a loss of density.
Reading between the lines
- The same reduction method should apply to other Furstenberg families defined by density or logarithmic density, giving matching Gδσ bounds and non-Fσδ examples wherever the relevant return-time sets form an analytic P-ideal; this is an extension the paper does not carry out.
- The printed gap in Claim 4, if repaired as the authors indicate, is unlikely to change the theorem's conclusion, but until the inequality is corrected a careful reader should treat the proof of Theorem 2.1 as conditional.
- The separation in Theorem 2.3 suggests that product-topology upper frequent hypercyclicity is a substantially weaker dynamical condition than its norm counterpart, which may matter for attempts to transfer frequent-hypercyclicity criteria from normed spaces to locally convex spaces.
- The paper's remark on weakly Farah ideals points to a natural strengthening: if every Fσδ-ideal belonged to the modified class Π̂03, then the Gδσ conclusion would extend beyond the analytic P-ideal case.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the descriptive complexity of the set UFHC(T) of upper frequently hypercyclic vectors of a continuous linear operator T on a second countable topological vector space. The first main result, Theorem 1.2, improves the known bound from G_{δσδ} to G_{δσ} for every analytic P-ideal I, by showing that HCT(I) is G_{δσ}. The second main result, Theorem 2.1, answers negatively a question of Bonilla and Grosse-Erdmann: for certain weighted backward shifts on ℓp with the product topology, UFHC(T) is not an F_{σδ}-set, hence not G_δ. The paper also contains Theorems 2.2 and 2.3 on the (non-)equivalence of norm, weak, and pointwise I-hypercyclicity for weighted backward shifts. The proofs combine a topological reduction to hereditary subfamilies of P(ω), a descriptive-set-theoretic transfer from a known non-F_{σδ} set in Baire space, and a Bayart–Ruzsa characterization of frequently hypercyclic weighted shifts.
Significance. If the proofs are corrected, the paper makes a solid contribution. Theorem 1.2 is a clean and elegant improvement of the known upper bound, and the explicit reduction in Theorem 3.6 is an original way to transfer a descriptive-set-theoretic lower bound to UFHC(T). The negative answer to Question 1.3 is a genuine advance, and Theorems 2.2 and 2.3 address a natural and interesting comparison between topologies. The paper is generally well written, with self-contained arguments for the main upper bound and a clear strategy for the lower bound. The main reservations concern two specific proof gaps, one of which is load-bearing for Theorem 2.1; both appear to be repairable within the scope of the manuscript.
major comments (2)
- [3.2 (Construction step (iii) and Claim 4, pp. 9–11)] The proof of Theorem 3.6 asserts in step (iii) that one may choose m_{t+1} such that, for all m ≥ m_{t+1} and all j ∈ [0,t+1], μ_{α_t+m}(S_j \ [0,α_t)) ≥ (1−ε_{t+1}) d⋆(S_j), and it states that this is possible because d⋆(S_j) > 0. This is false as written: positive upper density is only a limsup property, so good scales need not occur eventually. However, the hypothesis of Theorem 3.6 is pointwise frequent hypercyclicity, which by Definition 3.5 gives d_*(S_j) > 0, i.e. lower density is positive. Replacing d⋆ by d_* in step (iii) and in the corresponding display in Claim 4 makes the universal condition valid and the final lower bound positive. The additional inequality in Claim 4 where β_{t−1}+(x_t+1)hat m_t is replaced by (j+2)hat m_t is not a problem: since m_t > t^2 α_{t−1}, one has β_{t−1} ≤ t α_{t−1} < hat m_t, so the inequality has the correct direction. As printed, the proof of Theorem 3.6 is incomplete, and this gap is load-bearing for Theorem 2.1.
- [3.3 (Proof of Theorem 2.2, p. 12)] The paragraph beginning 'Observe that ∥T∥ > 1' contains a false implication. From |(T^n y)_0| ≥ 1/2 and (T^n y)_0 = w_1···w_n y_n it is concluded that |y_n| ≥ 1/2, but this requires |w_1···w_n| ≤ 1, which is not established and is generally false when ∥T∥ > 1. For example, with w_n = 2 and y_n = 2^{−n}, one has (T^n y)_0 = 1 for all n while y ∈ ℓp. This paragraph is not needed for the rest of the proof and should be removed or replaced by a correct argument; as it stands, it invalidates the printed proof of Theorem 2.2.
minor comments (4)
- [3.1 (Proof of Theorem 3.2, p. 7)] In the induction step for F ∈ \hatΠ^0_3, the text says HCT(F) = ∪_j HCT(F_j) ∈ Π^0_3(X); the conclusion should be Σ^0_3(X), consistent with the statement of Theorem 3.2(ii).
- [3.2 (Step (i) of the construction, p. 9)] The same d⋆ versus d_* issue appears in step (i), where the condition for m_0 is written with d⋆; after the correction to d_*, the condition becomes justified by the frequent-hypercyclicity hypothesis.
- [3.2 (Theorem 3.7, p. 12)] The name 'Bayart and Rusza' should be 'Bayart and Ruzsa', matching the reference list.
- [Definition 3.5 and throughout] The notation for lower density d_* and upper density d⋆ is easy to confuse in the typeset text; using a clearly distinct symbol for the lower density would prevent the kind of slip identified in the main comments.
Circularity Check
No significant circularity: the new complexity bounds are derived from external characterizations (Bayart–Ruzsa, Solecki) and elementary Borel-class transfers, not from the paper's own conclusions.
full rationale
Searching the derivation chain: Theorem 1.2 is proved from Proposition 3.1/Theorem 3.2 plus the external fact that analytic P-ideals lie in the modified class Pi-hat^0_3, cited to [18] (Hrusak-Meza-Alcantara) and based on Solecki's submeasure representation [27]; the inductive identities HCT(union F_j) = intersection HCT(F_j) and HCT(intersection F_j) = union HCT(F_j) are explicit and do not presuppose the conclusion. Theorem 2.1 reduces to Theorem 3.6, whose frequent-hypercyclicity hypothesis is supplied by Theorem 3.7, quoted from Bayart-Ruzsa [5, Theorem 4]: condition (2.1) is equivalent to norm frequent hypercyclicity. This is an external, parameter-free equivalence; the example is not obtained by fitting UFHC(Bw) itself. The self-citations [22], [23], [24] are background context or standard ideal-cluster-point facts and are not load-bearing premises of the new G_delta_sigma upper bound or of the non-F_sigma_delta lower bound. No uniqueness theorem from the authors' prior work is invoked, and no known empirical result is renamed. I do flag a separate proof defect in Theorem 3.6, step (iii) and Claim 4: positive upper density is only a limsup property, so the printed universal threshold 'for all m >= m_{t+1}' is not guaranteed; the construction is incomplete as written and appears repairable by choosing m_t for the finitely many required indices j. That is an error in the proof, not a circular dependence on the theorem's conclusion. Hence circularity score 0.
Assumptions & free parameters
assumptions (5)
- standard math Descriptive set theory facts: C3 = {lim x_n = ∞} is Fσδ but not Gδσ; ∆ is a Polish space; continuous preimages of Fσδ sets are Fσδ.
- standard math Mazur's characterization of Fσ ideals: I = Fin(φ) = {S : φ(S) < ∞} for a lower semicontinuous submeasure φ, with φ(ω) = ∞ (Eq (3.1), corrected).
- standard math Solecki's characterization: analytic P-ideals are of the form Exh(φ) and belong to the class Π̂0_3.
- domain assumption Bayart-Ruzsa theorem: for bounded positive weights on ℓp, frequent hypercyclicity, upper frequent hypercyclicity, and condition (2.1) are equivalent.
- standard math Countably generated ideals are Fσ, admit an lscsm representation, and are Q+ ideals.
Cite this review
Pith. "Pith review of On the complexity of upper frequently hypercyclic vectors." pith.science (2026). https://pith.science/paper/CAU32FEO
@misc{pith2026250622341,
author = {Pith},
title = {Pith review of: On the complexity of upper frequently hypercyclic vectors},
year = {2026},
howpublished = {\url{https://pith.science/paper/CAU32FEO}},
note = {Machine review of arXiv:2506.22341}
}
abstract
Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{UFHC}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x \in X$ such that $\{n \in \omega: T^nx \in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{UFHC}(T)$ is a $G_{\delta\sigma\delta}$-set which is either empty or contains a dense $G_{\delta}$-set. Using a purely topological proof, we improve it by showing that $\mathrm{UFHC}(T)$ is always a $G_{\delta\sigma}$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. \textbf{31} (2018), 673--711] whether $\mathrm{UFHC}(T)$ is always a $G_{\delta}$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{UFHC}(T)$ is not a $F_{\sigma\delta}$-set (hence not $G_\delta$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology.
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