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A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times

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Pith's one-line read Bilinear pointwise ergodic averages along integer parts of αn converge for every real α

desk verdict Serious new machinery for real-alpha floor sequences, but the rational-gamma reduction asserted in Section 1 is not just notational and the stated theorems are not proved as written. read the letter →

arxiv 2501.06877 v1 pith:Y5LOBV53 submitted 2025-01-12 math.DS math.CA

Ben Krause This is my paper · ORCID
classification math.DSmath.CA MSC 37A3037A4542B25
keywords pointwiseergodictheoremsdoublerecurrencereturntimestheorembilinearaveragesKroneckersequenceoscillationinequalitiestime-frequencyanalysismetricentropy
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

Bourgain's double-recurrence theorem says that for rational step sizes the averages of a product of two shifted functions converge pointwise; his return-times theorem says the same for averages weighted by an orbit. This paper proves both statements when the shifts are integer parts of $\alpha n$ for every real $\alpha$, and when the second shift is an integer part of $\gamma n$ for every rational $\gamma$. The bilinear averages converge almost everywhere on every $\sigma$-finite measure-preserving system, and in the return-times setting one full-measure set of starting points works simultaneously for all $\gamma\in\mathbb{Q}$ and all auxiliary systems. The interest is that the proof is quantitative and unified: it reduces both theorems to a uniform bound on an oscillation counting function, then analyzes that function with entropy-based stopping times and time-frequency orthogonality.

What carries the argument

The load-bearing mechanism is the oscillation counting function $C_{\tau,B,H}$, defined as the largest $K$ for which there exist $M_0<M_1<\cdots<M_K\le H/100$ with $\mu(\{x:\max_{M_{k-1}\le M\le M_k}|\Phi_M-\Phi_{M_k}|\ge\tau\})\ge\tau^B$; the paper proves Proposition 4.3, that $C_{\tau,B,H}\lesssim_{\tau,B}1$ uniformly in $H$ and in the system, and the analogous return-times version with the supremum over auxiliary systems finite $\nu$-a.e. The argument decomposes the averaged functions into Fourier multipliers $\Omega_{J,\delta}$ restricted to level sets, then organizes dyadic intervals into trees and branches via an energy pigeonholing 'Forest Lemma' (Lemma 8.5), whose exceptional-set estimate rests on the tile orthogonality bound of Proposition 6.4. Single-scale control is achieved through vector-valued entropy estimates (the jump-counting lemmas of Section 6.3), and the floor function is resolved by Poisson summation (Lemma 7.2). The resolution of measurability in the Appendix uses Halmos' weak-density theorem to reduce to a fixed ergodic system.

What would settle it

For the aperiodic but non-ergodic system $S(x,y)=(x+\alpha\bmod 1,y)$ on $\mathbb{T}\times\{0,1\}$ with $\nu=\mathrm{Leb}\times(\tfrac12,\tfrac12)$, compute the return-times oscillation count $C^{(X_0,\mu_0,T_0)}_{\tau,B,H}(\omega)$ for a fixed ergodic $(X_0,T_0)$ and a function $g$ depending on $y$; if this is not bounded uniformly in $H$ on a full-measure set of $\omega$, Theorem 1.4 collapses, and the failure would directly pinpoint Lemma 14.5's uniform approximation claim.

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Extended reading notes

Core claim

The paper's central claim is that for each fixed $\alpha\in\mathbb{R}$, the bilinear pointwise ergodic averages $\frac{1}{N}\sum_{n\le N}T^{\lfloor\alpha n\rfloor}f\cdot T^{\lfloor\gamma n\rfloor}g$ converge almost everywhere for every $\gamma\in\mathbb{Q}$ and every pair $f,g\in L^\infty$ on any $\sigma$-finite measure-preserving system, and that the analogous return-times averages $\frac{1}{N}\sum_{n\le N}T^{\lfloor\alpha n\rfloor}f\cdot S^{\lfloor\gamma n\rfloor}g(\omega)$ converge almost everywhere for $\nu$-a.e. $\omega$, simultaneously for every auxiliary system, with the convergence set independent of $\gamma$. The quantitative engine is a uniform bound on an oscillation counting function $C^{(X,\mu,T)}_{\tau,B,H}$ that is independent of the system and of $H$, plus a pointwise-in-$\omega$ version for return times. The proofs replace the irrational step $\alpha$ by a family of Fourier multipliers localized near level sets of the secondary function $g$, organized into trees and branches by an entropy-driven selection procedure, and then estimate maximal and oscillation operators by combining vector-valued martingale jump inequalities with a Bessel-type tile orthogonality estimate. Both theorems are recovered from the oscillation bounds via a Calderón transference to the integers and a brief contradiction argument.

Load-bearing premise

The load-bearing premise is that Halmos' weak-density theorem gives a uniform-in-H approximation of any aperiodic countably generated system by a conjugate of a fixed ergodic system with error $o(\tau^B H^{-100})$, in the exact quantifier order needed to transfer the return-times oscillation bound.

Editorial extensions

If this is right

  • For every $\alpha\in\mathbb{R}$ and $\gamma\in\mathbb{Q}$, the averages $\frac{1}{N}\sum_{n\le N}T^{\lfloor\alpha n\rfloor}f\,T^{\lfloor\gamma n\rfloor}g$ converge $\mu$-a.e. for all $f,g\in L^\infty$ on any $\sigma$-finite system, with the convergence set independent of $\gamma$.
  • The return-times averages converge for $\nu$-a.e. $\omega$ simultaneously over all auxiliary systems and all $\gamma\in\mathbb{Q}$, so a single full-measure set $Y_g$ works for every rational multiplier.
  • For $g=1_E$ with $S$ ergodic, the averages of $T^{\lfloor\alpha n\rfloor}f$ over the return times $\{n\le N:S^n\omega\in E\}$ converge pointwise for $\nu$-a.e. $\omega$ and $\mu$-a.e. $x$, for every real $\alpha$.
  • The proof yields uniform-in-$H$ oscillation bounds, so convergence holds without ergodicity of $T$ and without a dense subclass of model functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same tree-branch-entropy mechanism may extend to averages along $\lfloor p(n)\rfloor$ for integer-valued polynomials $p$ of degree $\ge 2$, but the Fourier localization step would need a more delicate equidistribution analysis than the paper gives for linear sequences.
  • The uniformity in $\gamma\in\mathbb{Q}$ suggests the return-times convergence set could be chosen independent of a continuum of multipliers $\gamma\in\mathbb{R}$; the paper's level-set decomposition would need to be uniform over $\mathbb{R}$, which is not established.
  • A natural testable strengthening is that the convergence set in Theorem 1.1 is independent of $\alpha$ as well, since the argument's constants depend on $\alpha$ only through the auxiliary integer $D_\alpha$; the paper does not claim this.
  • The quantitative oscillation bound $C_{\tau,B,H}\lesssim_{\tau,B}1$ should lead to explicit rates of convergence, which the original Bourgain arguments did not provide; the paper introduces the quantification but does not extract rates.
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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

3 major / 3 minor

Summary. The paper claims a unified proof of two pointwise ergodic theorems for averages sampled along the integer parts of linear sequences. Theorem 1.1 asserts that for every real α, every rational γ, every σ-finite measure-preserving system, and all bounded f,g, the averages (1/N)∑ T^{⌊αn⌋}f · T^{⌊γn⌋}g converge almost everywhere. Theorem 1.4 asserts the analogous return-times statement with a single full-measure set Y_g that works uniformly over all γ∈Q and all auxiliary systems. The proof is organized around quantifying oscillation, transferring to the integers, decomposing Fourier multipliers into forests and branches, and proving maximal and oscillation estimates; an appendix addresses measurability and reductions to ergodic systems.

Significance. If correct, the result is substantial: it extends Bourgain's double-recurrence and return-times theorems from the case α∈Q to all real α and to all rational γ, with convergence sets independent of γ. The paper's architecture is coherent and ambitious, and its quantitative formulation in Proposition 4.3 is a genuine strength, as is the explicit transference framework of §5 and the forest/branch machinery of §8. The author is also honest about several dependencies: Proposition 6.4 is imported from Thiele's book, Lemma 14.5 is only sketched, and the reduction from γ∈Q to γ=−1 is asserted rather than derived. These gaps are load-bearing, so the paper cannot be accepted in its present form, but they are plausibly fixable within the manuscript's scope.

major comments (3)
  1. [§1 and §7–§13] Theorem 1.1 and Theorem 1.4 quantify over every γ∈Q, but the proof after §1 is carried out only for γ=−1. The sentence in §1 that the general case involves 'only notational' changes after splitting into residue classes is not supported by a derivation. Writing γ=p/q and n=qk+r gives floor terms floor(αqk+αr) for the first factor and floor(−pk−r) for the second; the first sequence has a nonzero additive constant, and the second has step −p, not −k. None of Lemma 7.2, the multiplier organization of §8, or the maximal/oscillation estimates of §§10–13 is shown to be invariant under these operations. As written, the proof does not establish the theorem even for γ=2, and it certainly does not establish the stated uniform-in-γ convergence sets. This reduction must be proved or the theorems must be restricted accordingly.
  2. [§6, Proposition 6.4 and Lemma 8.5] The Forest Lemma 8.5, which is central to the exceptional-set estimate (8.6), relies on the tile orthogonality estimate Proposition 6.4. That proposition is stated but its proof is only described as 'identical' to [32, Lemma 5.1], with a decomposition of φ sketched in two lines. Since this estimate is the quantitative engine behind the entire forest decomposition and is used repeatedly in the subsequent sections, the paper should give either a complete proof or a precise statement of the cited lemma together with a verification that all hypotheses of the present setting are met. As it stands, a key input is a black box.
  3. [§14, Lemma 14.5] The proof of Lemma 14.5 is explicitly only a sketch, but Lemma 14.5 is needed to reduce the return-times theorem from an arbitrary aperiodic countably generated system (Y,ν,S) to an ergodic one. The return-times conclusion requires uniform control in H and in the auxiliary system (X,µ,T), and the sketeched weak-density approximation does not demonstrate the correct quantifier order: the exceptional set for the approximating ergodic system must be controlled independently of H and uniformly over all auxiliary systems. Without a complete proof of this reduction, Theorem 1.4 is not established for the full class of systems stated in the theorem.
minor comments (3)
  1. [Theorem 1.4] There is a typo in the statement: 'for for all g ∈ L∞(Y)' should read 'for all g ∈ L∞(Y)'.
  2. [§14, Lemma 14.2] Lemma 14.2 introduces a countably generated assumption on (X,µ) only in a footnote-like parenthetical, but Theorems 1.1 and 1.4 are stated for arbitrary σ-finite systems. Since f and g generate a countably generated sub-σ-algebra under the action of T, this is likely fixable, but the reduction should be stated explicitly.
  3. [§9, footnote 2] The footnote says that the case α∈Q is simpler and reduces to α∈Z, but no details are given. Since the main theorems include rational α, at least a brief indication of this reduction should appear in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorems are parameter-free convergence claims proved from external harmonic-analytic and ergodic-theoretic tools, with no fitted input renamed as a prediction.

full rationale

The paper's central claims, Theorems 1.1 and 1.4, are parameter-free almost-everywhere convergence statements about bilinear ergodic averages along floor(Kronecker) sequences. Nothing in the proof is fitted to a subset of the target data and then relabeled as a prediction: the quantitative oscillation and maximal estimates of Propositions 4.3, 9.1, and 9.2 are universal bounds with explicit dependence only on the fixed parameters tau, B, H, delta, and alpha. The derivation chain passes through Calderon transference, Fourier-multiplier organization, tree-branch selection, entropy/jump estimates, and the Forest Lemma; these are standard external tools imported from Bourgain, Thiele, Demeter, and Halmos, not from an unverified self-citation chain. The author's self-citations, e.g. [19] for a standard decomposition into indicator functions in the appendix and [20] for expository discussion, are not load-bearing: the relevant argument is supplied in the appendix and the cited lemmas are routine. Two passages warrant scrutiny but do not constitute circularity: the assertion that reduction from gamma in Q to gamma = -1 involves 'only notational' changes after residue-class splitting, and the 'Sketch' proof of Lemma 14.5 reducing to ergodic (Y, nu, S). If either reduction fails, the result would be a correctness gap, not a circular derivation. There is no equation in the paper that equals its own input by construction, no fitted parameter renamed as a convergence set, and no uniqueness theorem invoked from the author's prior work to force the chosen framework. The proof is self-contained relative to the cited external machinery, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data-fitted or empirical constants appear in the theorem. The many symbols in Section 2.2 (A0, B_tau, R, W, t, rho, Delta, V, epsilon, etc.) are internal proof parameters bounded by powers of tau and delta; the theorem statement itself is parameter-free. No new particles, forces, or entities are introduced.

assumptions (6)
  • standard math Birkhoff pointwise ergodic theorem and weak-type (1,1) maximal ergodic inequality for measure-preserving transformations.
    Invoked in Section 5 for transference and Pointwise Ergodic Theorem, and in Section 14 for dense subclass reduction.
  • standard math Calderon transference principle for translation-invariant operators.
    Used in Sections 1 and 5 to reduce ergodic averages to discrete integer averages.
  • standard math Halmos' theorem that conjugates of an aperiodic transformation are weakly dense among measure-preserving transformations.
    Used in Appendix Section 14, Lemmas 14.4 and 14.5, to replace arbitrary and non-ergodic systems by fixed ergodic ones.
  • standard math Time-frequency Bessel orthogonality estimate, Proposition 6.4, taken from Thiele's book [32, Lemma 5.1].
    Load-bearing for the Forest Lemma (Lemma 8.5) and the tree and branch decomposition; cited without full proof in the preprint.
  • standard math Vector-valued jump-counting and variational inequalities for dyadic martingales (Lemmas 6.6, 6.8, 6.11) and their corollaries.
    Quoted from prior literature, including the author's book [19], and used to control entropy and oscillation.
  • domain assumption Reduction to countably generated Lebesgue probability spaces and ergodic systems for the return-times theorem.
    The return-times theorem is stated for aperiodic countably generated systems; the proof in Section 14 further reduces to ergodic Lebesgue spaces via Halmos' theorem.

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Pith. "Pith review of A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times." pith.science (2026). https://pith.science/paper/Y5LOBV53

@misc{pith2026250106877,
  author       = {Pith},
  title        = {Pith review of: A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5LOBV53}},
  note         = {Machine review of arXiv:2501.06877}
}
abstract

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the following two results for each $\alpha \in \mathbb{R}$: First, for each $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and each $f,g \in L^{\infty}(X)$, for each $\gamma \in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor } f \cdot T^{\lfloor \gamma n \rfloor} g \] converge $\mu$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,\nu,S)$, and each $g \in L^{\infty}(Y)$, there exists a subset $Y_{g} \subset Y$ with $\nu(Y_{g})= 1$ so that for all $\gamma \in \mathbb{Q}$ and $\omega \in Y_{g}$, for any auxiliary $\sigma$-finite measure-preserving system $(X,\mu,T)$, and any $f \in L^{\infty}(X)$, the ``return-times" averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor} f \cdot S^{\lfloor \gamma n \rfloor } g(\omega) \] converge $\mu$-a.e. Moreover, in both cases the sets of convergence are identical for all $\gamma \in \mathbb{Q}$.

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