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arxiv: 2511.02080 · v2 · pith:ZMFTFVEVnew · submitted 2025-11-03 · 🧮 math.DS

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

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

Pith reviewed 2026-05-18 00:45 UTC · model grok-4.3

classification 🧮 math.DS
keywords return-time setsminimal dynamical systemspronilfactorsmultiple recurrencecharacteristic factorspolynomial sequencessyndetic setstopological dynamics
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The pith

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

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

The paper proves a structure theorem for return-time sets generated by polynomial tuples in minimal topological dynamical systems. These sets match the corresponding sets in the system's maximal infinite-step pronilfactor except for differences that are not piecewise syndetic. The argument uses characteristic factor theory to reduce questions about the original system to its pronilfactor. The result produces new multiple recurrence theorems and shows that two earlier conjectures are equivalent.

Core claim

In a minimal topological dynamical system, the return-time set determined by a polynomial tuple along an orbit coincides, up to a non-piecewise syndetic set, with the corresponding return-time set in the system's maximal infinite-step pronilfactor.

What carries the argument

The maximal infinite-step pronilfactor of the minimal system, which captures the polynomial recurrence information after reduction via characteristic factor theory.

If this is right

  • Three new multiple recurrence theorems hold for linear recurrence along dynamically defined syndetic sets.
  • Polynomial recurrence along arithmetic progressions occurs in both minimal and totally minimal systems.
  • The Glasner-Huang-Shao-Weiss-Ye conjecture and the Leibman conjecture are equivalent.

Where Pith is reading between the lines

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

  • The equivalence of the two conjectures means that a proof of one immediately yields a proof of the other through this structure theorem.
  • The reduction to pronilfactors may allow similar structural results for return-time sets along other sequences beyond polynomials.
  • Explicit computations in low-complexity minimal systems such as irrational rotations could provide direct checks on the size of the exceptional non-piecewise syndetic sets.

Load-bearing premise

Topological characteristic factor theory reduces the minimal system to its maximal pronilfactor while preserving the structure of the polynomial return-time sets.

What would settle it

A concrete minimal system and polynomial tuple where the return-time set differs from the one in its maximal pronilfactor by a piecewise syndetic set would falsify the structure theorem.

read the original abstract

We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, 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. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a structure theorem for polynomial return-time sets in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, it shows that for a minimal system the return-time sets along polynomial tuples coincide, up to a non-piecewise syndetic exceptional set, with the corresponding sets in the system's maximal infinite-step pronilfactor. Applications include three new multiple recurrence theorems (linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems) and a proof that two previously separate conjectures (one of Glasner-Huang-Shao-Weiss-Ye and one of Leibman) are equivalent.

Significance. If the result holds, the structure theorem supplies a clean reduction principle that transfers questions about polynomial return times in arbitrary minimal systems to the pronilfactor setting, where polynomial recurrence is already well-understood. The exceptional-set control is stated in terms of non-piecewise syndeticity, preserving compatibility with the combinatorial notions already used in the return-time literature. The applications yield concrete new recurrence statements and establish an equivalence between two lines of conjectures, thereby unifying previously distinct strands of work in topological dynamics.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'three new multiple recurrence theorems' is stated without even a one-sentence indication of their content; a brief parenthetical description would make the abstract self-contained.
  2. [Introduction / §2] The manuscript invokes the Glasner-Huang-Shao-Weiss-Ye characteristic-factor theorem without citing a specific numbered statement from that reference; adding the precise theorem label would help readers locate the exact reduction used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting the significance of the structure theorem and its applications, and for recommending acceptance. We are pleased that the reduction principle to the pronilfactor and the unification of the two conjectures were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives its structure theorem for polynomial return-time sets by invoking the existing topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye (distinct authors with no overlap to the current team) to reduce to the maximal infinite-step pronilfactor, then applying known polynomial recurrence results there. This is standard citation of independent prior work rather than self-citation load-bearing, self-definition, or renaming of fitted inputs. No equations or steps in the provided abstract or outline reduce the central claim to its own inputs by construction; the exceptional-set control is compatible with existing syndeticity notions without circular reduction. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the topological characteristic factor theory of Glasner et al. and standard background assumptions of compactness and continuity in topological dynamics; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye
    Invoked to reduce minimal systems to their maximal infinite-step pronilfactors.

pith-pipeline@v0.9.0 · 5673 in / 1129 out tokens · 33535 ms · 2026-05-18T00:45:18.660303+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

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    Algebraic characterization of RP^[d] via new topology and proof that order d-1 maximal factors are topological characteristic factors for higher-order configurations in group actions.

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

    math.DS 2026-01 unverdicted novelty 7.0

    U^k(Φ)-uniform sets contain rich families of infinite sumsets whose structure scales with k, subject to higher-order parity obstructions coming from nilsystems.

Reference graph

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