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$\mathrm{IP}_{\mathrm{rat}}$-polynomial recurrence and large intersections

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Pith's one-line read Rationally independent nonlinear polynomials produce large intersections along every rational-spectrum IP set.

desk verdict Solid upgrade of Frantzikinakis–Kra large intersections to positive lower IP_rat density for nonlinear independent polynomials, with a clean new pointwise IP_rat formula on nilmanifolds. read the letter →

arxiv 2607.09358 v1 pith:DTXBLBPD submitted 2026-07-10 math.DS math.CO

Borys Holikov , Or Shalom This is my paper · ORCID
classification math.DSmath.CO MSC 37A3037A0511B3028D05
keywords IPsetsrationalspectrumpolynomialrecurrencelargeintersectionscharacteristicfactorsHost–KraKroneckerfactornilmanifolds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that if you take a set of natural numbers that is dense in the usual sense, and you look for simultaneous returns along several rationally independent nonlinear polynomials, those returns are not only positive but nearly as large as independent chance would predict, and they occur with positive relative density inside every IP set generated by a sequence whose spectrum is rational. The same statement holds in the dynamical setting: for any measure-preserving system and any positive-measure set A, the multiple intersections A ∩ T^{p1(n)}A ∩ … ∩ T^{pk(n)}A stay larger than µ(A)^{k+1}-ε on a set of n that has positive lower IP_rat-density. The authors reach this by first showing that Host–Kra factors control the polynomial averages taken along rational IPs, then refining the characteristic factor all the way down to the rational Kronecker factor, and finally obtaining a pointwise limit formula for continuous functions on nilmanifolds that generalizes Leibman’s classical result.

What carries the argument

IP_rat-characteristic factors: a factor Y of a system X is IP_rat-characteristic for a polynomial system P if the multiple ergodic averages along any rational-spectrum IP vanish whenever one of the functions is orthogonal to Y. The paper proves that Host–Kra factors of finite order, and ultimately the rational Kronecker factor itself, are IP_rat-characteristic for nonlinear rationally independent polynomials.

What would settle it

Exhibit a single linear polynomial together with one or more nonlinear polynomials that remain rationally independent, and a system in which the corresponding multiple averages along some rational-spectrum IP do not vanish when a function orthogonal to every Host–Kra factor is inserted; this would refute the natural extension of the characteristic-factor theorem.

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Extended reading notes

Core claim

For every invertible measure-preserving system, every positive-measure set A, every family of rationally independent nonlinear integer polynomials p1,…,pk vanishing at zero, and every ε>0, the set of n for which µ(A ∩ T^{p1(n)}A ∩ … ∩ T^{pk(n)}A) exceeds µ(A)^{k+1}-ε has positive lower IP_rat-density. The combinatorial translation is that the same large-intersection set intersects every IP generated by a rational-spectrum sequence with positive relative density along every Følner sequence.

Load-bearing premise

The reduction that produces a standard polynomial family and identifies the rational Kronecker factor as characteristic works only when every polynomial is strictly nonlinear; the presence of even one linear polynomial breaks the argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves that for any invertible measure-preserving system, any set A of positive measure, and any family of rationally independent nonlinear integer polynomials p1,...,pk with pi(0)=0, the set of n for which the multiple intersection measure exceeds µ(A)^{k+1}-ε has positive lower IP_rat-density (Theorem 1.7). The combinatorial counterpart (Corollary 1.8) follows by the Furstenberg correspondence principle. The argument proceeds by establishing that Host–Kra factors Z_{k-1}(X) are IP_rat-characteristic for standard nonlinear polynomial systems (Proposition 3.8 via PET induction and an IP van der Corput lemma), reducing general nonlinear systems to the standard case (Theorem 3.9), proving a pointwise limit formula for continuous functions on nilmanifolds along rational IPs (Theorems 4.5 and 5.9, generalizing Leibman), and finally identifying the rational Kronecker factor as the minimal IP_rat-characteristic factor (Theorem 6.2). The large-intersection claim is then obtained by a weighted averaging argument that exploits the positivity of the density of n with all polynomial values congruent to 0 mod r (Lemma 7.2).

Significance. The result simultaneously upgrades the Frantzikinakis–Kra large-intersection theorem for polynomials from syndeticity to positive lower IP_rat-density and extends the linear IP_rat results of Kra–Shalom to the nonlinear setting. The pointwise convergence theorem for polynomial sequences along rational IPs (Theorem 5.9) is of independent interest and supplies a clean generalization of Leibman’s classical formula. Within the stated nonlinear regime the proofs are complete and rest on standard, non-circular black boxes (Host–Kra structure, Leibman equidistribution, the authors’ earlier IP mean-ergodic theorem). The restriction to strictly nonlinear polynomials is flagged explicitly and converted into a precise conjecture (Conjecture 1.9), so the paper does not overclaim.

minor comments (5)
  1. Abstract and title use “IPrat” / “IP_rat” inconsistently with the body; a single notation should be fixed throughout.
  2. Definition 2.4 of IP as a multiset is non-standard; a short remark clarifying that multiplicities do not affect the averages would help readers.
  3. In the proof of Lemma 3.5 the appeal to Sperner’s theorem is correct but could be replaced by a simpler double-counting argument; either way a one-line reference or expansion would improve readability.
  4. Page 17, display after “we have that H=(G°)^k”: the product of integrals is written with connected components X_{x_j,i}; a brief reminder that these are the components of the original system would avoid a momentary ambiguity.
  5. Several arXiv preprints are cited as 2025/2026; once published versions appear the bibliography should be updated.

Circularity Check

1 steps flagged · score 1.0 of 10

Minor self-citation of the IP_rat mean-ergodic theorem and van der Corput lemma from prior joint work; the polynomial large-intersection claim is derived independently via PET induction and nilmanifold equidistribution adaptations.

  1. self citation load bearing [Section 2.3, Theorem 2.11 and Lemma 2.13]
    "The following mean ergodic theorem along IP-iterates was established in [28]. ... Another important result of [28] is the following adaptation of van der Corput lemma for IP averages."

    These two results (IP mean ergodic theorem and IP van der Corput) from the second author's prior paper with Kra are invoked as black boxes for every subsequent IP_rat average, the PET induction in Proposition 3.8/Theorem 3.9, and the pointwise nilmanifold formulae. They are load-bearing for the method, yet they establish only the linear/IP infrastructure and do not encode or assume the nonlinear polynomial large-intersection property proved here; the circularity is therefore only the formal self-citation of the averaging toolkit, not a reduction of Theorem 1.7 to itself.

full rationale

The paper's derivation of Theorem 1.7 (and Corollary 1.8) proceeds by establishing that Host–Kra factors are IP_rat-characteristic for nonlinear rationally independent polynomials (Theorem 3.9, via PET induction on weights using the IP van der Corput lemma), then strengthening to the rational Kronecker factor (Theorem 6.2, via Leibman-style pointwise limits on nilmanifolds and Frantzikinakis–Kra equidistribution of diagonal orbits), and finally transferring large intersections via approximation by finite rational Kronecker factors and a positive-density argument for the indicator ψ_r^P. All steps are self-contained mathematical reductions under the stated nonlinear hypothesis; the only external inputs are classical structure theorems (Host–Kra, Leibman, Frantzikinakis–Kra) and the authors' prior IP framework (mean ergodic theorem and van der Corput for rational-spectrum IPs). Those prior results supply the averaging infrastructure but do not contain or force the target large-intersection statement for polynomials, so the self-citation is ordinary tool reuse rather than circular reduction of the claim to its own inputs. No self-definitional loops, fitted parameters, uniqueness smuggling, or renaming of known empirical patterns appear. Score is therefore 1 (minor non-load-bearing self-citation of framework).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely inside classical ergodic theory and additive combinatorics. No free parameters are fitted. The only non-standard axioms are the existence of Host–Kra factors and the IP mean-ergodic theorem for rational spectrum, both taken from the literature. No new physical or combinatorial entities are postulated.

assumptions (4)
  • domain assumption Host–Kra structure theorem: for an ergodic system the factors Z_k(X) are inverse limits of k-step nilsystems and are characteristic for ordinary polynomial averages.
    Invoked throughout §3 and §6 to reduce multiple averages to nilsystems (Theorem 2.3).
  • domain assumption IP mean-ergodic theorem for sequences with rational spectrum (Kra–Shalom).
    Theorem 2.11 supplies existence of the IP averages used in every subsequent limit.
  • domain assumption Leibman’s equidistribution and pointwise convergence results for polynomial sequences on nilmanifolds.
    Used as the base case that is extended to IP_rat averages in §4–5.
  • standard math Standard measure-preserving systems, Følner sequences, and the Furstenberg correspondence principle.
    Background language of the entire paper.

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Pith. "Pith review of $\mathrm{IP}_{\mathrm{rat}}$-polynomial recurrence and large intersections." pith.science (2026). https://pith.science/paper/DTXBLBPD

@misc{pith2026260709358,
  author       = {Pith},
  title        = {Pith review of: $\mathrmIP_\mathrmrat$-polynomial recurrence and large intersections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DTXBLBPD}},
  note         = {Machine review of arXiv:2607.09358}
}
abstract

Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $\Phi$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_{\Phi}\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right) > d_{\Phi}(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.

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