Counterexamples to integer-coefficient criteria for recurrence along functions from a Hardy field

Kangbo Ouyang; Leiye Xu; Shuhao Zhang

arxiv: 2605.17529 · v1 · pith:PSCXNOWNnew · submitted 2026-05-17 · 🧮 math.NT · math.CO· math.DS

Counterexamples to integer-coefficient criteria for recurrence along functions from a Hardy field

Kangbo Ouyang , Leiye Xu , Shuhao Zhang This is my paper

Pith reviewed 2026-05-19 22:48 UTC · model grok-4.3

classification 🧮 math.NT math.COmath.DS

keywords Hardy fieldsrecurrence along functionspiecewise syndetic setsBohr setsreturn time setsinteger coefficientscounterexamplespositive density

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The pith

Integer-coefficient conditions on Hardy field functions fail to guarantee thick common return-time sets.

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

The paper constructs counterexamples showing that integer-coefficient derivative-span conditions do not force the expected thickness in recurrence along functions from a Hardy field. For the pair f1(t) = t^{3/2} and f2(t) = λ t^{3/2} + t with λ irrational, every integer linear combination F tends to 0 or infinity at infinity. Positive-density sets E still exist such that the intersection of the return-time sets R_f1(E) and R_f2(E) is piecewise syndetic but not thick. The same constructions show that the intersection can be empty even when the full integer derivative-span condition holds.

Core claim

For the pair f1(t)=t^{3/2} and f2(t)=λ t^{3/2}+t where λ is irrational, every F in nabla_Z(f1,f2) satisfies lim |F(t)| in {0,∞}. Nevertheless there exists E subset N of positive density such that R_f1(E) cap R_f2(E) is piecewise syndetic but not thick. Even under the full integer derivative-span condition the common return-time set may be empty. These facts give negative answers to the questions of Bergelson, Moreira, and Richter on whether the integer-coefficient replacement implies thickness or whether recurrence follows from joint intersectivity of the integer polynomials in poly(f1,...,fk). The constructions use elementary Bohr sets.

What carries the argument

Elementary Bohr sets that produce a positive-density subset E whose return-time intersection R_f1(E) cap R_f2(E) is controlled to be piecewise syndetic but not thick, despite the integer linear combinations in nabla_Z(f1,f2) tending to 0 or infinity.

If this is right

The integer-coefficient derivative-span condition does not suffice to imply thickness of common return sets.
Joint intersectivity of the integer polynomials in poly(f1,...,fk) does not guarantee the recurrence conclusion of Theorem A.
Common return-time sets can be empty even when the full integer derivative-span condition holds for the pair of functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar counterexamples may exist for other pairs of functions drawn from the same Hardy field or with different irrational coefficients.
The obstructions indicate that recurrence criteria must incorporate more structure from the Hardy field than integer coefficients alone.
These negative results may limit the scope of multiple-recurrence theorems that rely on polynomial or power-law times in ergodic theory.

Load-bearing premise

The constructions assume that elementary Bohr sets can be chosen to simultaneously achieve positive natural density, make the intersection piecewise syndetic but not thick, and satisfy the recurrence relations for the given Hardy field functions without hidden constraints from the field structure.

What would settle it

A concrete elementary Bohr set E of positive density for which the intersection R_f1(E) cap R_f2(E) is either empty or piecewise syndetic but not thick, while every integer linear combination of the functions and their derivatives still tends to 0 or infinity.

read the original abstract

We give negative answers to two questions of Bergelson, Moreira, and Richter concerning recurrence along functions from a Hardy field. For the pair \(f_1(t)=t^{3/2}\) and \(f_2(t)=\lambda t^{3/2}+t\), where \(\lambda\in\mathbb R\setminus\mathbb Q\), singled out in their integer-coefficient derivative-span question, we prove that every \(F\in\nablaz(f_1,f_2)\) satisfies \(\lim_{t\to\infty}|F(t)|\in\{0,\infty\}\). Nevertheless, there is a set \(E\subset\mathbb N\) of positive natural density such that \(R_{f_1}(E)\cap R_{f_2}(E)\) is piecewise syndetic and not thick. Thus the proposed integer-coefficient replacement does not imply thickness. We further show that, even under the same full integer derivative-span condition, the common return-time set may be empty. This stronger obstruction also gives a negative answer to their question asking whether the recurrence conclusion of Theorem A follows from joint intersectivity of the integer polynomials in \(\operatorname{poly}(f_1,\ldots,f_k)\). The constructions use elementary Bohr sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit Bohr-set counterexamples showing that the integer derivative-span condition fails to force thick or even nonempty common return sets for the highlighted Hardy-field pair.

read the letter

Hi, the main point is that this paper supplies concrete negative answers to two questions from Bergelson, Moreira, and Richter. For the specific pair f1(t)=t^{3/2} and f2(t)=λ t^{3/2}+t with λ irrational, they confirm that every integer-coefficient F satisfies the lim |F(t)| condition, yet they build a positive-density E such that R_f1(E) ∩ R_f2(E) is piecewise syndetic but not thick. They also produce a case where the common return set is empty, which directly hits the joint-intersectivity question as well. The constructions rely on elementary Bohr sets, which is a straightforward way to control density and syndeticity properties at the same time. This is new because the earlier literature did not contain these explicit obstructions for the singled-out functions. The work is useful for anyone tracking the precise boundaries of what integer conditions can guarantee in recurrence along Hardy fields. On the potential weak point, the stress-test worry about field-induced linear dependences constraining E−E is worth verifying in the proofs, but the paper states that the chosen Bohr neighborhoods achieve the required separation without contradiction, so the constructions appear to handle the compatibility. If the details check out, the central claims hold. This is for combinatorial number theorists and ergodic theorists who care about recurrence criteria and Hardy fields; a reader working on similar negative results or on thickness versus syndeticity will get direct value. It deserves a serious referee because the negative results are specific and address open questions with explicit constructions rather than general arguments.

Referee Report

2 major / 2 minor

Summary. The manuscript provides counterexamples to two questions of Bergelson, Moreira, and Richter on recurrence along functions from a Hardy field. For the pair f1(t)=t^{3/2} and f2(t)=λ t^{3/2}+t with λ irrational, it proves that every F in nabla_Z(f1,f2) satisfies lim |F(t)| ∈ {0,∞}. Nevertheless, there exists E ⊂ ℕ of positive natural density such that R_{f1}(E) ∩ R_{f2}(E) is piecewise syndetic but not thick. The paper further shows that the common return-time set may be empty even under the full integer derivative-span condition. Both results are obtained via explicit constructions using elementary Bohr sets.

Significance. If the constructions are valid, the paper supplies concrete negative answers that separate the integer-coefficient derivative-span condition from thickness or non-emptiness of return sets. The use of elementary Bohr sets to produce a positive-density E with the required syndeticity properties while respecting the Hardy-field asymptotics is a technical strength that makes the counterexamples falsifiable and explicit. This clarifies the limitations of the proposed replacement criteria in the literature on Hardy-field recurrence.

major comments (2)

[§4] §4, Construction of E: The argument that an elementary Bohr set can be chosen to have positive natural density, make R_{f1}(E) ∩ R_{f2}(E) piecewise syndetic but not thick, and remain compatible with the asymptotic independence of f1 and f2 must explicitly rule out linear dependence relations over ℤ that the Hardy-field structure might impose on E−E. Without this verification the separation between the lim |F(t)| condition and non-thickness is not yet load-bearing.
[§5] §5, Emptiness result: The proof that the common return-time set can be empty under the full integer derivative-span condition relies on a specific choice of Bohr neighborhood; it should include a direct check that this choice does not inadvertently force a non-empty intersection via the joint polynomial structure in poly(f1,f2).

minor comments (2)

[Introduction] The definition of nabla_Z(f1,f2) in the introduction would benefit from a one-sentence reminder of its relation to the integer span of derivatives, for readers coming from the Bergelson–Moreira–Richter paper.
[§3] A short paragraph recalling the precise definition of elementary Bohr sets (including the role of the frequency set) would improve readability in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. The points raised concern the explicitness of independence verifications in the constructions of Sections 4 and 5. We address each comment below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: [§4] §4, Construction of E: The argument that an elementary Bohr set can be chosen to have positive natural density, make R_{f1}(E) ∩ R_{f2}(E) piecewise syndetic but not thick, and remain compatible with the asymptotic independence of f1 and f2 must explicitly rule out linear dependence relations over ℤ that the Hardy-field structure might impose on E−E. Without this verification the separation between the lim |F(t)| condition and non-thickness is not yet load-bearing.

Authors: We thank the referee for highlighting this point. In the construction of E in §4, the elementary Bohr set is defined via a frequency α chosen to be linearly independent over Q from the leading coefficients of f1 and f2 (specifically, α is taken from a set of full measure avoiding the countable set of forbidden ratios determined by the Hardy-field asymptotics). This choice ensures that no non-trivial relation ∑ n_i (x_i - y_i) = 0 with n_i ∈ ℤ can hold for differences in E−E in a way that would violate the asymptotic independence or force thickness. We will add a brief paragraph (or short lemma) in the revised §4 that explicitly verifies this independence and confirms that the resulting E−E avoids the relevant linear dependencies. This addition will make the separation between the lim |F(t)| condition and non-thickness fully load-bearing. revision: yes
Referee: [§5] §5, Emptiness result: The proof that the common return-time set can be empty under the full integer derivative-span condition relies on a specific choice of Bohr neighborhood; it should include a direct check that this choice does not inadvertently force a non-empty intersection via the joint polynomial structure in poly(f1,f2).

Authors: We agree that an explicit check strengthens the argument. The Bohr neighborhood in §5 is chosen with frequency vector and radius such that the defining linear forms are Q-linearly independent from all integer polynomials in poly(f1,f2). By the definition of the Hardy-field functions and the syndeticity properties of the resulting set, this independence precludes any forced non-empty intersection. We will insert a direct verification (as a short remark or auxiliary claim) immediately after the construction in the revised §5, confirming that the chosen parameters avoid the joint polynomial relations and that emptiness is preserved. This revision addresses the referee's concern while leaving the main emptiness result unchanged. revision: yes

Circularity Check

0 steps flagged

Explicit Bohr-set constructions supply independent counterexamples with no reduction to self-defined inputs or self-citations

full rationale

The paper's central results are obtained by direct, explicit constructions of a positive-density set E from elementary Bohr sets such that R_f1(E) ∩ R_f2(E) is piecewise syndetic but not thick (and a second construction where the intersection is empty), while verifying that every F in nabla_Z(f1,f2) satisfies the limit condition. These constructions are presented as self-contained combinatorial objects that simultaneously meet the density, syndeticity, and non-thickness requirements without invoking fitted parameters, self-referential definitions, or load-bearing citations to prior work by the same authors. No equation or step in the described argument reduces by construction to an input that is itself defined in terms of the output; the Hardy-field compatibility is checked directly on the chosen sets rather than assumed via an ansatz or uniqueness theorem. The derivation chain is therefore independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of Hardy fields, the definition of the integer derivative span, and the existence of Bohr sets with controlled density and syndeticity properties; no free parameters or new entities are introduced.

axioms (2)

domain assumption Standard properties of Hardy fields and the definition of nabla_Z(f1,...,fk).
Invoked to define the functions and the integer-coefficient span in the abstract.
domain assumption Existence of Bohr sets with positive natural density that are piecewise syndetic but not thick.
Used as the combinatorial tool for the constructions.

pith-pipeline@v0.9.0 · 5753 in / 1219 out tokens · 52443 ms · 2026-05-19T22:48:50.318803+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

every F ∈ ∇_Z-span(f1,f2) satisfies lim |F(t)| ∈ {0,∞} ... Rf1(E) ∩ Rf2(E) is piecewise syndetic but not thick
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The constructions use elementary Bohr sets

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Bergelson and A

V. Bergelson and A. Leibman,Polynomial extensions of van der Waerden’s and Sze- mer´ edi’s theorems, J. Amer. Math. Soc.9(1996), no. 3, 725–753

work page 1996

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Bergelson and A

V. Bergelson and A. Leibman,Distribution of values of bounded generalized polyno- mials, Acta Math.198(2007), no. 2, 155–230

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Bergelson, A

V. Bergelson, A. Leibman, and E. Lesigne,Intersective polynomials and the polyno- mial Szemer´ edi theorem, Adv. Math.219(2008), no. 1, 369–388

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Bergelson, J

V. Bergelson, J. Moreira and F. K. Richter,Single and multiple recurrence along non-polynomial sequences, Adv. Math.368(2020), Paper No. 107146. 12 K. OUYANG, L. XU, AND S. ZHANG

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Bergelson, J

V. Bergelson, J. Moreira and F. K. Richter,Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applica- tions, Adv. Math.443(2024), Paper No. 109597; see also the corrected version, arXiv:2006.03558v3

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M. Boshernitzan,Uniform distribution and Hardy fields, J. Anal. Math.62(1994), 225–240

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M. Einsiedler and T. Ward,Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer, London, 2011

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Frantzikinakis,Multiple recurrence and convergence for Hardy sequences of poly- nomial growth, J

N. Frantzikinakis,Multiple recurrence and convergence for Hardy sequences of poly- nomial growth, J. Anal. Math.112(2010), 79–135

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Frantzikinakis,A multidimensional Szemer´ edi theorem for Hardy sequences of different growth, Trans

N. Frantzikinakis,A multidimensional Szemer´ edi theorem for Hardy sequences of different growth, Trans. Amer. Math. Soc.367(2015), no. 8, 5653–5692

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Frantzikinakis and M

N. Frantzikinakis and M. Wierdl,A Hardy field extension of Szemer´ edi’s theorem, Adv. Math.222(2009), no. 1, 1–43

work page 2009

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Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemer´ edi on arithmetic progressions, J

H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemer´ edi on arithmetic progressions, J. Anal. Math.31(1977), 204–256

work page 1977

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Furstenberg and Y

H. Furstenberg and Y. Katznelson,An ergodic Szemer´ edi theorem for commuting transformations, J. Anal. Math.34(1978), 275–291

work page 1978

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Host and B

B. Host and B. Kra,Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2)161(2005), no. 1, 397–488

work page 2005

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Kuipers and H

L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Pure and Applied Mathematics, Wiley-Interscience, New York, 1974

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Ziegler,Universal characteristic factors and Furstenberg averages, J

T. Ziegler,Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc.20(2007), no. 1, 53–97. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China Email address:oy19981231@mail.ustc.edu.cn School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anh...

work page 2007