Pith Number

pith:ZMFTFVEV

pith:2025:ZMFTFVEVNQLPIRXUBVEMBJB43J

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A structure theorem for polynomial return-time sets in minimal systems

Andreas Koutsogiannis, Anh N. Le, Daniel Glasscock, Donald Robertson, Florian K. Richter, Joel Moreira

In minimal systems, polynomial return-time sets coincide with those in the maximal infinite-step pronilfactor up to non-piecewise syndetic sets.

arxiv:2511.02080 v2 · 2025-11-03 · math.DS

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Claims

C1strongest claim

We prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor.

C2weakest assumption

The argument depends on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye to reduce the system to its pronilfactor.

C3one line summary

In minimal systems, polynomial return-time sets coincide up to non-piecewise syndetic sets with those in the maximal infinite-step pronilfactor, yielding new multiple recurrence theorems.

References

32 extracted · 32 resolved · 0 Pith anchors

[1] J. Auslander. Minimal flows and their extensions , volume 153 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1988. Notas de Matem´ atica [Mathematical Notes], 122. 9, 1 1988

[2] K. Berg. Quasi-disjointness in ergodic theory. Trans. Amer. Math. Soc., 162:71–87, 1971. 15 1971

[3] V. Bergelson, B. Host, and B. Kra. Multiple recurrence and nilsequences. Invent. Math. , 160(2):261–303, 2005. With an appendix by Imre Ruzsa. 29 2005

[4] V. Bergelson, A. Leibman, and E. Lesigne. Intersective polynomials and the polynomial Szemer´ edi theorem. Adv. Math., 219(1):369–388, 2008. 23 2008

[5] V. Bergelson and R. McCutcheon. An ergodic IP polynomial Szemer´ edi theorem. Mem. Amer. Math. Soc., 146(695):viii+106, 2000. 3, 4, 13 2000

Formal links

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Cited by

2 papers in Pith

Infinite sumsets in $U^k(\Phi)$-uniform sets

On higher order regionally proximal relations and topological characteristic factors for group actions

Receipt and verification

First computed	2026-05-18T03:10:11.897841Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cb0b32d4956c16f446f40d48c0a43cda5b421c4a5e10cd32378db583a823eb3e

Aliases

arxiv: 2511.02080 · arxiv_version: 2511.02080v2 · doi: 10.48550/arxiv.2511.02080 · pith_short_12: ZMFTFVEVNQLP · pith_short_16: ZMFTFVEVNQLPIRXU · pith_short_8: ZMFTFVEV

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZMFTFVEVNQLPIRXUBVEMBJB43J \
  | 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: cb0b32d4956c16f446f40d48c0a43cda5b421c4a5e10cd32378db583a823eb3e

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "d0cac82708ee385cd2e624d401612bed7c16c93eb32734fbff4a8366328a09c4",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2025-11-03T21:40:30Z",
    "title_canon_sha256": "f803ddb8cb6363a0c4c5e6bc9620caa6339de156861785fc643a432626d285b6"
  },
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  "source": {
    "id": "2511.02080",
    "kind": "arxiv",
    "version": 2
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