{"id":"5f09820f-2c2a-44d8-a7b8-3bd283f09d74","arxiv_id":"2501.06877","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every real alpha and rational gamma, averaging with powers T^{floor(alpha n)} and T^{floor(gamma n)} converges almost everywhere, and the return-times analogue holds for aperiodic systems.","lead":"This paper proves that two Bourgain-type pointwise ergodic theorems, double recurrence and return times, hold when the steps are the integer parts of alpha times n. A single oscillation-and-entropy argument covers every real alpha and every rational gamma, with the same convergence set for all gamma.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed reduction from γ∈Q to γ=-1 is not merely notational: for rational γ, residue splitting produces floor terms with additive constants and two distinct integer steps that the γ=-1 estimates as written do not cover.","rationale":"I read the manuscript in good faith. The Fourier-analytic core is substantial, and the γ=-1 case may well be correct; the Forest Lemma and maximal/oscillation estimates are arranged coherently, and the cited black-box estimates are standard. My concern is with the unsupported bridge from γ=-1 to every γ∈Q. That bridge is more load-bearing than the Appendix 14 measurability/ergodic-reduction issue identified by the reader, because it concerns the theorem's stated quantifier over γ: if the residue-splitting reduction fails, the proof does not even reach Bourgain's original γ=2 case. The reader's conditional verdict is appropriate, but their weakest_assumption is not the decisive gap in my reading. The concrete test I propose would settle whether the reduction is genuinely notational or hides a new floor-phase and multi-step phenomenon; until then the central claim as stated is not fully established.","tokens_in":46381,"tokens_out":27198,"duration_ms":285333,"concrete_test":"For α=1 and γ=2, write out the claimed residue-class reduction explicitly and check whether the resulting averages can be rewritten in the A_M form of §5 with a single real slope and no additive floor constant. Equivalently, re-run Lemma 7.2 with w(αn+c-m) in place of w(αn-m) and verify whether an e(cξ) phase factor appears; if it does, test whether the estimates of §§8-13 remain uniform in c. If either step fails, the reduction from γ∈Q to γ=-1 is not notational and Theorem 1.1 is unproven for γ=2.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 1 asserts that it suffices to prove Theorem 1.1 for γ=-1, since the general γ∈Q case involves 'only notational' changes after splitting into residue classes and exploiting periodicity. This reduction is load-bearing because the abstract and Theorem 1.4 quantify over every γ∈Q. Write γ=p/q and n=qk+r. The average becomes a sum over r of terms (1/K)∑_{k≤K} T^{⌊αqk+αr⌋}f · T^{-pk-r}g. Two independent obstructions appear. First, the first sequence is floor(βk+c) with c=αr≠0; this is not the sequence floor(αn) analyzed in Lemma 7.2 and §9, and floor(p²x)≠p²floor(x), so the additive constant cannot be removed by rescaling the index. Second, the second step is -pk, not -k; reducing it to the analyzed case would require either a p-th root of T or an additional residue split that again introduces a nonzero additive floor constant. The paper supplies no derivation that the Poisson summation identity (Lemma 7.2), the multiplier organization (§8), or the oscillation and maximal estimates (§§10-13) are unaffected by these operations. Consequently, as written, the proof does not establish the theorem for γ=2, Bourgain's original double-recurrence exponent, let alone for all rational γ.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46656,"tokens_out":3327,"duration_ms":37502,"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":[{"comment":"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.","section":"§1 and §7–§13"},{"comment":"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.","section":"§6, Proposition 6.4 and Lemma 8.5"},{"comment":"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.","section":"§14, Lemma 14.5"}],"minor_comments":[{"comment":"There is a typo in the statement: 'for for all g ∈ L∞(Y)' should read 'for all g ∈ L∞(Y)'.","section":"Theorem 1.4"},{"comment":"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.","section":"§14, Lemma 14.2"},{"comment":"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.","section":"§9, footnote 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is very long and technically demanding, and the referee's confidence in the main result is conditional on filling the γ-reduction and Appendix gaps. The paper's core strategy appears sound and the quantitative formulation is valuable, but the stated theorems are stronger than what the written proof supports. I recommend major revision rather than rejection, because the missing pieces seem repairable within the manuscript's framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Krause has written a serious, long proof that aims to put the floor sequence floor(αn), α∈R, into Bourgain's pointwise bilinear theory. For the γ=-1 case the machinery is substantial: the transference to Z, the tree/branch decomposition, the entropy jump-counting, and the single-scale estimates are all present in real detail, and I did not find a contradiction in the parts that are fully written. A unified proof of double recurrence and return times by one argument is a genuine methodological contribution, and the real-alpha extension is a real open problem before this paper.\n\nBut the paper as stated is not proved. Section 1 asserts that after splitting into residue classes, the general rational γ reduces 'only notational' to γ=-1. That is simply not true. For γ=p/q, write n=qk+r. The two sequences become floor(βk+c) with β=αq and c=αr, and pk+d, respectively. Neither appears anywhere in the proof. The Poisson-summation identity in Lemma 7.2, the multiplier organization in §8, and the oscillation/maximal estimates in §§10–13 are all developed for floor(αm) paired with -m. No derivation is given for shifted floors or for the p-step. This is not a matter of adding constants: floor(βk+c) does not equal floor(βk)+c, and the second step would require a p-th root of T, which arbitrary measure-preserving systems do not admit. So as written, the theorems do not cover γ=2, let alone all of Q. That is a load-bearing gap.\n\nThere are smaller issues worth noting: Proposition 6.4 is a black box from Thiele's book, and Lemma 14.5 is explicitly a sketch. Those are etiquette problems for a paper of this length. The appendix's uniform-in-H weak-density argument is delicate, but I would not call it obviously wrong.\n\nBottom line: the γ=-1 core may well be right, and the paper deserves a serious referee. But the referee's first job is to demand a real proof of the rational-γ reduction; without it, Theorems 1.1 and 1.4 are unproven.","headline":"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.","tokens_in":47172,"tokens_out":6673,"would_cite":false,"duration_ms":65736,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A45","42B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bilinear pointwise ergodic averages along integer parts of αn converge for every real α","keywords":["pointwise ergodic theorems","double recurrence","return times theorem","bilinear ergodic averages","Kronecker sequence","oscillation inequalities","time-frequency analysis","metric entropy"],"falsifier":"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.","tokens_in":46163,"feed_emoji":"🔁","tokens_out":14141,"duration_ms":79904,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two Bourgain pointwise ergodic theorems now hold for all real α","feed_subtitle":"A unified proof gives almost everywhere convergence for both theorems, with one convergence set for all rational γ.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes Bourgain's double recurrence theorem for rational α; the paper extends this to all real α and adapts its hard-analysis template.","marker":"[7]"},{"why":"Proves pointwise ergodic theorems along arithmetic sets, including linear averages along ⌊αn⌋ and the return-times theorem in an appendix for rational α; supplies the multi-frequency maximal inequality.","marker":"[6]"},{"why":"Bourgain's unpublished manuscript that essentially proves the α∈Q case of the return-times theorem by harmonic analysis, which the present paper extends to real α.","marker":"[5]"},{"why":"Thiele's book supplies Proposition 6.4, the Bessel-type tile orthogonality estimate that underpins the Forest Lemma's packing bound.","marker":"[32]"},{"why":"Demeter's unpublished note is the source of the modern time-frequency and entropy approach to Bourgain's double recurrence that the paper reorganizes.","marker":"[9]"},{"why":"Calderón's transference principle converts the ergodic-theoretic oscillation problem into a discrete problem on the integers.","marker":"[8]"},{"why":"Halmos' weak-density theorem is used in the appendix to reduce arbitrary and non-ergodic systems to a fixed ergodic system, the step that carries the return-times uniformity.","marker":"[14]"},{"why":"Provides the vector-valued jump-counting estimates and the indicator-reduction for measurability that control the entropy-based selection of trees and branches.","marker":"[19]"}],"fun_headline_variants":["Bourgain's two theorems now hold for every real α","Unified proof: Bourgain double recurrence and return times for all α","All real α: two Bourgain pointwise ergodic theorems","Both Bourgain theorems extended to integer parts of any real step","Pointwise convergence for all real α in both Bourgain theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bourgain's two theorems now hold for every real α","Unified proof: Bourgain double recurrence and return times for all α","All real α: two Bourgain pointwise ergodic theorems","Both Bourgain theorems extended to integer parts of any real step","Pointwise convergence for all real α in both Bourgain theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000494,"raw_usage":{"total_tokens":2538,"prompt_tokens":1169,"completion_tokens":1369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":785,"completion_tokens_details":{"reasoning_tokens":1281}},"tokens_in":785,"tokens_out":1369,"duration_ms":9491,"temperature":1.0,"reasoning_tokens":1281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:50:01.669378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bourgain","cited_arxiv_id":null,"evidence_quote":"Establishes Bourgain's double recurrence theorem for rational α; the paper extends this to all real α and adapts its hard-analysis template."},{"cited_title":"Bourgain","cited_arxiv_id":null,"evidence_quote":"Proves pointwise ergodic theorems along arithmetic sets, including linear averages along ⌊αn⌋ and the return-times theorem in an appendix for rational α; supplies the multi-frequency maximal inequality."},{"cited_title":"Bourgain","cited_arxiv_id":null,"evidence_quote":"Bourgain's unpublished manuscript that essentially proves the α∈Q case of the return-times theorem by harmonic analysis, which the present paper extends to real α."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Thiele's book supplies Proposition 6.4, the Bessel-type tile orthogonality estimate that underpins the Forest Lemma's packing bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demeter's unpublished note is the source of the modern time-frequency and entropy approach to Bourgain's double recurrence that the paper reorganizes."},{"cited_title":"Calder´ on","cited_arxiv_id":null,"evidence_quote":"Calderón's transference principle converts the ergodic-theoretic oscillation problem into a discrete problem on the integers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Halmos' weak-density theorem is used in the appendix to reduce arbitrary and non-ergodic systems to a fixed ergodic system, the step that carries the return-times uniformity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the vector-valued jump-counting estimates and the indicator-reduction for measurability that control the entropy-based selection of trees and branches."}],"review_version":1}