Pith Number

pith:PSCXNOWN

pith:2026:PSCXNOWNNCXAO2VKCZKAFIQVAO

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

Kangbo Ouyang, Leiye Xu, Shuhao Zhang

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

arxiv:2605.17529 v1 · 2026-05-17 · math.NT · math.CO · math.DS

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Claims

C1strongest 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,∞}, yet 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.

C2weakest assumption

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.

C3one line summary

Counterexamples demonstrate that integer-coefficient derivative-span conditions fail to imply thickness or non-emptiness of common return-time sets for recurrence along Hardy field functions.

References

15 extracted · 15 resolved · 0 Pith anchors

[1] 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 1996

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

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

[4] 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 2020

[5] 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 2024

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:44.224356Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7c8576bacd68ae076aaa165402a21503a0f9cbd7dde3e0d45761312f95684360

Aliases

arxiv: 2605.17529 · arxiv_version: 2605.17529v1 · doi: 10.48550/arxiv.2605.17529 · pith_short_12: PSCXNOWNNCXA · pith_short_16: PSCXNOWNNCXAO2VK · pith_short_8: PSCXNOWN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PSCXNOWNNCXAO2VKCZKAFIQVAO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7c8576bacd68ae076aaa165402a21503a0f9cbd7dde3e0d45761312f95684360

Canonical record JSON

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    "abstract_canon_sha256": "f9ce8328df0b9455387b0d1edb9d77662405faefbbf6cdc3e3985278af5488bf",
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    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-17T16:26:45Z",
    "title_canon_sha256": "2d082696a5ad1feaf047ba4dc11580df499983a765d75f5980d60c92796d1433"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17529",
    "kind": "arxiv",
    "version": 1
  }
}