{"id":"0f00407a-b16e-4fe2-8a47-6a1cb342824a","arxiv_id":"1908.02281","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Claims almost-everywhere convergence of bilinear ergodic averages along arbitrary polynomials and along primes, but the proof omits the key rotated polynomial oscillation estimates.","lead":"This paper claims a simplified proof of Bourgain's bilinear ergodic theorem and extends it to averages along polynomials and along primes. The proof relies on oscillation estimates, but the key polynomial estimates are stated without proof.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof does not establish Theorem 2.1 for general P,Q: the uniform-in-θ rotated oscillation estimate (8.3)-(8.4) is asserted without proof, and the interpolation in Theorem 2.2 does not follow from the endpoints derived.","rationale":"The reader's verdict of REJECT is supported. The strongest_claim is Theorem 2.1, and the proof fails exactly at the point where distinct polynomials P and Q are introduced. Section 8 merely states the rotated oscillation estimates (8.3)-(8.4) with a reference to the simple spectrum of the shift, but no argument is supplied; this is the load-bearing bridge from the proved symmetric case Q=-P (Equation (7.5)) to the general case. Lemmas 6 and 7 themselves are quoted from the literature for unweighted polynomial averages; the uniform-in-θ weighted versions are new, and the paper's own text shows no derivation. The passage to L^r×L^{r'} is also not justified: the maximal inequality in Theorem 2.2 is intended to enable a density argument, but its proof derives an endpoint that does not interpolate to the claimed range. The final remark about a 'careful bilinear interpolation' with range 1≤1/r+1/r'<3/2 further indicates the interpolation step was not carried out. These are internal proof gaps, not disagreements with external consensus. The manuscript may describe a plausible strategy, but as written it establishes the full theorem only in a special case; the reader's rejection is appropriate. No change to the verdict is needed.","tokens_in":21222,"tokens_out":8137,"duration_ms":85769,"concrete_test":"Independently re-derive (8.3) for a concrete case, e.g., R(n)=n^2, Q(n)=n, from the circle-method estimates cited for Lemma 6 (Bourgain [6,7]; Nair [33, Lemmas 4-5]), tracking the phase θ through the Weyl sums. If the cited estimates give uniform control only when the phase and shift polynomials are equal up to sign (the Q=-P case), then (8.3) is not a consequence of Lemma 6, and the reduction to arbitrary P,Q in Theorem 2.1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires a.e. convergence for arbitrary nonconstant integer-valued polynomials P,Q and f∈L^r, g∈L^{r'} with 1/r+1/r'=1. The only step addressing P≠-Q is Section 8, where (8.3)-(8.4) assert a 'rotated version' of Lemmas 6 and 7: for every θ, oscillation estimates for averages weighted by e^{iR(n)θ} along Q(n), uniformly in θ, with no proof. This is not a minor omission: it is precisely the estimate needed in (7.6)-(7.7) to control the inner integral after the Fourier identity. Lemmas 6 and 7, as stated and referenced to Bourgain [6,7] and Nair [33], control unweighted averages along a single polynomial; inserting an oscillatory factor e^{iR(n)θ} with a second polynomial changes the underlying Weyl sums, and uniformity in θ is a new quantitative statement. The application at (7.7) treats the modulated average g_θ as though Lemma 6 applied directly, which is the same gap. Additionally, the proof of Theorem 2.2 does not yield the claimed maximal inequality: (8.2) proves an ℓ1×ℓr → ℓr bound, and with the trivial ℓ∞×ℓ∞ → ℓ∞ bound the claimed bilinear interpolation would need to produce ℓr×ℓr' → ℓ1, which the stated endpoints do not give; no ℓ1×ℓ∞ → ℓ1 endpoint is proved. Without this maximal inequality, the extension from L∞ to L^r×L^{r'} is unsupported. The concluding remark admits the interpolation is only sketched and proposes a range 1≤1/r+1/r'<3/2, which is not the same as the conjugate-exponent range stated in Theorem 2.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to give a simple proof of Birkhoff's ergodic theorem, a proof of Bourgain's homogeneous bilinear ergodic theorem, and an extension of these results to averages along arbitrary non-constant integer-valued polynomials and along primes evaluated at polynomials. The main theorem (Theorem 2.1) asserts almost everywhere convergence of (1/N)Σ f(T^{P(n)}x)g(T^{Q(n)}x) and the prime analogue for f∈L^r, g∈L^{r'} with 1/r+1/r'=1 for any invertible measure-preserving T. The proof strategy follows Bourgain's oscillation method: a spectral identity reduces the bilinear average to an integral of a linear average of a rotated function g_θ; oscillation lemmas (Lemmas 6 and 7, borrowed from Bourgain and Nair) are then invoked; and Theorem 2.2 claims a bilinear maximal inequality that is used to pass from L^∞ to L^r×L^{r'}. Section 7 derives the convergence only in the special case Q = -P (Equation (7.5)), while Section 8 asserts, without proof, a 'rotated version' of Lemmas 6 and 7 (equations (8.3)-(8.4)) that would handle general polynomial pairs, and also claims a bilinear interpolation step that is not carried out.","tokens_in":21663,"tokens_out":1986,"duration_ms":20320,"significance":"If the main theorem were fully proved, it would be a significant contribution: it would generalize Bourgain's bilinear ergodic theorem to polynomial iterates and to primes along polynomials, with a claimed sharp L^r×L^{r'} range. The paper also presents an expository proof of Birkhoff's theorem via Bourgain's oscillation method and cites several machine-checkable or reproducible lemmas from the literature. However, the central new claims are not established: the key estimates for general polynomial pairs are asserted rather than proved, and the interpolation argument does not mathematically close. The paper's strengths are its expository portions and the recognition of the role of the shift's simple Lebesgue spectrum, but the novelty of the extension is unsupported.","major_comments":[{"comment":"The proof of convergence in Section 7 is carried out only for the special case Q = -P. The identity (7.5) expresses the average along P(n) and -P(n), and the subsequent estimates (7.6)-(7.8) concern g(x-P(n)). Theorem 2.1 is stated for arbitrary non-constant integer-valued polynomials P and Q; the passage from Q = -P to general P,Q is not present in Section 7 and is deferred to the assertions in Section 8. This is a load-bearing gap for the main theorem.","section":"Section 7, Equation (7.5)"},{"comment":"The 'rotated version' of Lemmas 6 and 7 is asserted with no proof and no reference. The statements (8.3)-(8.4) claim a uniform-in-θ oscillation estimate for averages weighted by e^{iR(n)θ} along a polynomial Q, and the same for prime averages. This is precisely the estimate needed to control the integral in (7.6) when the two polynomials P and Q are distinct: the phase e^{iR(n)θ} arises from the second polynomial and changes the Weyl sums fundamentally. Bourgain's lemmas, as cited in [6,7,33], do not contain such a uniform oscillatory estimate. The phrase 'it can be seeing that' is not a proof. Since (8.3)-(8.4) is the only bridge from the proved case Q = -P to arbitrary P,Q, the main theorem for general polynomial pairs is unsupported.","section":"Section 8, Equations (8.3)-(8.4)"},{"comment":"The proof of Theorem 2.2 does not yield the claimed maximal inequality. Equation (8.2) establishes an ℓ1×ℓr → ℓr bound, and the text notes the trivial ℓ∞×ℓ∞ → ℓ∞ bound. The claimed conclusion for f∈L^r, g∈L^{r'} with 1/r+1/r'=1 would require an endpoint such as ℓ1×ℓ∞ → ℓ1 or an interpolation that produces ℓr×ℓ^{r'} → ℓ1. The stated endpoints do not give this: bilinear interpolation between ℓ1×ℓr → ℓr and ℓ∞×ℓ∞ → ℓ∞ yields bounds with norms on the right in a range that does not include the conjugate-exponent pair claimed in Theorem 2.1. The remark at the end of Section 8 concedes that the claimed range is 1 ≤ 1/r+1/r' < 3/2, which is not the same as the conjugate-exponent range in Theorem 2.1. Thus the maximal inequality, and hence the L^r×L^{r'} extension, is not established.","section":"Section 8, Equation (8.2) and 'bilinear interpolation'"},{"comment":"The proof of the prime version of Theorem 2.2 and Theorem 2.1 is not written out. The text says 'Applying the same machinery' and 'The case r = +∞ can be handled in the same manner', but no prime analogue of (8.1)-(8.2) is proved. Since the prime maximal inequality (7.4) is cited from Nair and the rotated prime estimate (8.4) is asserted without proof, the prime part of the main theorem inherits the same gaps and adds no independent derivation.","section":"Section 8, interpolated application to primes"},{"comment":"In the proof of Bourgain's bilinear ergodic theorem, the passage from the ℓ2(Z) estimate (6.5)-(6.6) to the L^2(X) estimate (6.7) is only sketched as 'using carefully similar arguments to that in the proof of Propositions 1 and 2'. Since the transference principle in Propositions 1 and 2 is stated for the homogeneous averages with the form f(T^n x)g(T^{-n}x), and the oscillation estimate requires control along a sequence of intervals, the transference of the oscillation norm is not immediate. This leaves the proof of the classical Bourgain theorem itself incomplete, although this portion is not the main novelty.","section":"Section 6, Equation (6.7)"}],"minor_comments":[{"comment":"There are numerous typographical errors and notation inconsistencies (e.g., 'Garcia' for 'Garsia', 'Etamedi' for 'Etemadi', 'curial' for 'crucial', 'Cauchy-Cauchy' repetitions, inconsistent use of [−π,π) and [−1/N,1/N] in Lemma 4 and its proof). These should be corrected.","section":"Throughout"},{"comment":"The displayed norm in Theorem 2.2 contains a misprint: it writes ||f||_r ||f||_{r'} instead of ||f||_r ||g||_{r'}. This should be fixed.","section":"Section 2, Theorem 2.2 statement"},{"comment":"Lemma 4 is stated for g positive integrable on the circle, while the proof of Lemma 5 appears to prove the stronger estimate for arbitrary θ with a uniform constant; the relationship between the two lemmas and the role of the spectral measure σ_f (which is not necessarily absolutely continuous with respect to Lebesgue measure) should be clarified. The estimate is used at a point where absolute continuity of σ_f is asserted without proof.","section":"Section 5, Lemma 4 and Lemma 5"},{"comment":"In the proof of Theorem 5.2, the spectral transfer uses σ_f and claims its absolute continuity with respect to Lebesgue measure; this is not generally true for arbitrary f∈ℓ2(Z). The argument appears to rely instead on the fact that the shift has Lebesgue spectrum, but the exposition is not precise about how σ_f is handled.","section":"Section 5, Theorem 5.1 vs Theorem 5.2"},{"comment":"The reference [2] is cited as the author's own work, but the connection to the present proof is not explained. Several references (e.g., [13], [37]) are cited for standard lemmas; a precise pointer to the claimed rotated estimates (8.3)-(8.4) would be essential, but none is given.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper's title and abstract promise a simple proof of a substantial theorem, but the proof of the main extension rests on two unproved assertions: the uniform-in-θ rotated oscillation inequalities (8.3)-(8.4) and a bilinear interpolation step that is not valid from the stated endpoints. The manuscript also has significant expository gaps in the proof of the classical Bourgain theorem itself. These are not merely presentation issues; they are load-bearing for the central claim. The paper may contain useful expository ideas, but in its current form it does not meet the standard for publication in a serious journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nWhat you should know about this paper: the headline theorem is unproven as written, and the two gaps are load-bearing, not cosmetic. But the paper is not a crackpot effort — there is a real and mostly correct account of Bourgain's bilinear ergodic theorem in the linear case, and the polynomial problem it targets is a genuine open extension. The issue is that the extension is exactly where the proof stops.\n\nWhat is new: the combination of two arbitrary non-constant polynomials with bilinear averages (and the prime version) is not in Bourgain or Nair, so if it worked, Theorem 2.1 would be a real advance. The early sections on the oscillation method and the spectral proof of Birkhoff are standard but reorganized cleanly, and the identity (7.5) for the symmetric case Q = -P is correct. The proof of the bilinear theorem for the symmetric linear case follows Bourgain's Fourier argument and seems essentially sound for L², with standard transference to a.e. convergence.\n\nSoft spots, in order. First, Section 8 asserts the rotated version of Lemmas 6 and 7, (8.3)-(8.4), for averages with an exponential weight e^{iR(n)θ}, uniformly in θ, with no proof or reference. This is the only bridge from the proved symmetric case to arbitrary Q ≠ -P, and it is a genuinely new quantitative statement — inserting an oscillatory factor into polynomial/prime Weyl sums is not a freebie. Second, the bilinear interpolation in Theorem 2.2 does not produce the claimed ℓ^r × ℓ^{r'} → ℓ^1 maximal inequality. What is actually proved in (8.2) is an ℓ^1 × ℓ^r → ℓ^r bound; combined with ℓ^∞×ℓ^∞→ℓ^∞ this interpolates to ℓ^p × ℓ^{rp} → ℓ^{rp}, which is not the conjugate-exponent result. The concluding remark's range 1 ≤ 1/r+1/r' < 3/2 is also not the same as Theorem 2.1's conjugate pairs. Third, even the symmetric polynomial case applies Lemma 6 to Q = -P, but Lemma 6 is stated for polynomials mapping naturals to themselves; negative-leading-coefficient P is not covered without an argument, though this is probably fixable by passing to T^{-1} or restricting to finite initial terms.\n\nVerdict: as submitted, Theorem 2.1 is not proven. The work may be salvageable if (8.3)-(8.4) are true and can be cited from Bourgain/Nair or proved, and if the interpolation is replaced with a correct multi-parameter argument. For a specialist who knows the Bourgain circle method, the paper is useful as a roadmap, but the main claim should not be accepted on this text. I would send it to a serious referee — the problem is real and the author is not a novice — but the referee's main job would be to test whether those two estimates exist in the literature or fail.\n\nRecommendation: engage with it as a preprint, treat Theorem 2.1 as open, and if you cite it, cite it as claimed rather than proved.","headline":"The claimed polynomial bilinear theorem is not established: the rotated oscillation estimates are asserted without proof and the interpolation step is flawed, though the linear Bourgain proof is mostly sound.","tokens_in":22118,"tokens_out":9123,"would_cite":false,"duration_ms":81231,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","28D05","05D10","11B30","11N37","37A45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that bilinear ergodic averages along two arbitrary integer-valued polynomials, and also along prime-indexed polynomial steps, converge almost everywhere.","keywords":["bilinear ergodic averages","polynomial ergodic averages","prime numbers","almost everywhere convergence","oscillation method","maximal inequalities","transference principle","spectral isomorphism"],"falsifier":"Choose $R(n)=n^2$, $Q(n)=n$, and for growing $K$ compute, on finite subsets of $\\mathbb Z$, the quantity $\\sum_{k=1}^K \\|\\sup_{N_k\\le N\\le N_{k+1}, N\\in S_\\rho} |\\frac{1}{N}\\sum_{n\\le N} e^{in^2\\theta}g(x+n)-\\frac{1}{N_{k+1}}\\sum_{n\\le N_{k+1}} e^{in^2\\theta}g(x+n)|\\|_{\\ell^2(\\mathbb Z)}$. If there are sequences $g\\in\\ell^2(\\mathbb Z)$ and phases $\\theta$ for which this grows like $\\sqrt{K}\\|g\\|_2$ rather than $o(\\sqrt{K})\\|g\\|_2$, the uniform rotated inequality (8.3) fails, and with it the paper's route from the proved case $Q=-P$ to arbitrary polynomial pairs.","tokens_in":21023,"feed_emoji":"📈","tokens_out":17262,"duration_ms":157939,"temperature":0.7,"pith_summary":"This paper gives a proof that bilinear ergodic averages taken along two arbitrary non-constant integer-valued polynomials $P,Q$ converge almost everywhere, both for ordinary time steps and for prime steps. In full, for any invertible measure-preserving transformation $T$ and any $f\\in L^r$, $g\\in L^{r'}$ with $\\frac{1}{r}+\\frac{1}{r'}=1$, the averages $\\frac{1}{N}\\sum_{n=1}^N f(T^{P(n)}x)g(T^{Q(n)}x)$ and their prime-indexed versions converge for almost every $x$. These are nonconventional ergodic averages, so the result is a pointwise, polynomial version of double recurrence. The proof is a modern simple proof based on the simple Lebesgue spectrum of the integer shift, a Fourier identity that rewrites the bilinear averages as integrals of linear averages of rotated functions, and maximal and oscillation estimates for polynomial and prime-polynomial averages; a maximal inequality for the supremum of these averages is isolated as a result of independent interest.","feed_headline":"Bilinear ergodic averages along polynomials and primes converge","feed_subtitle":"Extends the bilinear ergodic theorem to polynomial steps and prime-indexed polynomial steps.","key_machinery":"The load-bearing object is the identity (7.5), combined with the oscillation method. The identity is $$\\frac{1}{N}\\sum_{n=1}^N f(x+P(n))g(x-P(n)) = \\int_{-\\pi}^{\\pi} \\hat f(\\$\\theta$)\\left(\\frac{1}{N}\\sum_{n=1}^N g_\\$\\theta$(x-P(n))\\right)$e^{{2ix\\theta}}$\\,d\\$\\theta$,$$ with $g_\\theta(x)=g(x)e^{ix\\theta}$. It turns a product of two orbit segments into a continuum of ordinary linear averages of a single rotated function, so estimates from the linear theory can be integrated against $|\\hat f|$. The oscillation method is the scheme of Lemma 3 by which a bound on the sum over dyadic blocks of the $\\ell^2$ size of the maximal oscillation between block endpoints forces almost-everywhere convergence; the polynomial and prime-polynomial versions of these oscillation estimates are Lemmas 6 and 7, whose proofs rest on the discrete maximal inequality and the circle method for exponential sums. A spectral fact about the integer shift — that it has simple Lebesgue spectrum — is what, in this argument, upgrades these estimates to the rotated, uniformly-in-$\\theta$ form (8.3)-(8.4) needed for arbitrary $P,Q$.","core_discovery":"The central claim is Theorem 2.1: for any non-constant integer-valued polynomials $P(n),Q(n)$ and any invertible measure-preserving transformation $T$ of a probability space, the averages $\\frac{1}{N}\\sum_{n=1}^N f(T^{P(n)}x)g(T^{Q(n)}x)$ and $\\frac{1}{\\pi_N}\\sum_{p\\le N} f(T^{P(p)}x)g(T^{Q(p)}x)$ converge almost everywhere whenever $\\frac{1}{r}+\\frac{1}{r'}=1$ and $f\\in L^r$, $g\\in L^{r'}$. The discovery is that this family of results can be reached from a single oscillation scheme: prove the convergence on the integer lattice $\\ell^2(\\mathbb Z)$ via Fourier analysis and maximal inequalities, then transfer to an arbitrary dynamical system by a bilinear transference principle. The decisive mechanism is the identity (7.5), which rewrites the bilinear average $f(x+P(n))g(x-P(n))$ as an integral of linear averages of rotated functions $g_\\theta(x)=g(x)e^{ix\\theta}$; this reduces the bilinear problem to a family of linear polynomial averages. To pass from the symmetric case $Q=-P$, where (7.5) applies directly, to arbitrary pairs $P,Q$, the paper states a rotated version of its polynomial and prime-polynomial oscillation estimates, uniformly in the rotation parameter $\\theta$ (equations (8.3)-(8.4)).","pith_inferences":["If the rotated oscillation inequalities (8.3)-(8.4) are supplied with proof, the same Fourier-rotation scheme would plausibly extend to pairs of sequences whose exponential sums obey uniform Weyl-type estimates, including multi-dimensional $\\mathbb Z^d$ actions, not just polynomial powers of one transformation.","A finite-range numerical check of the uniform-in-$\\theta$ oscillation bound for, say, $R(n)=n^2$ and $Q(n)=n$ would provide evidence about whether the asserted rotated estimates are plausible; such a check cannot prove almost-everywhere convergence, but a clear violation would show the gap at (8.3)-(8.4) is genuine.","The paper's final conjecture about averages with operators in the weak closure of $T$ would become a natural target if the rotated estimates are established, since the same integral-of-linear-averages mechanism could then handle a much wider class of exponent sequences."],"forward_implications":["For any non-constant integer-valued polynomials $P,Q$, the averages $\\frac{1}{N}\\sum_{n=1}^N f(T^{P(n)}x)g(T^{Q(n)}x)$ converge for almost every $x$, for all $f\\in L^r$ and $g\\in L^{r'}$ with $\\frac{1}{r}+\\frac{1}{r'}=1$.","The same almost-everywhere convergence holds when the time parameter runs over primes: $\\frac{1}{\\pi_N}\\sum_{p\\le N} f(T^{P(p)}x)g(T^{Q(p)}x)$ converges for almost every $x$.","Theorem 2.2 yields a strong maximal inequality for the supremum in $N$ of the polynomial bilinear averages, with $L^1$ norm bounded by $C_r\\|f\\|_r\\|g\\|_{r'}$; this is the mechanism that passes from dense classes of functions to all admissible $L^r\\times L^{r'}$ pairs.","The same oscillation method gives a unified treatment of Birkhoff's theorem, the homogeneous bilinear ergodic theorem, and their polynomial and prime-polynomial extensions, with a bilinear transference principle carrying the integer-lattice estimates to arbitrary measure-preserving systems."],"supporting_citations":[{"why":"Supplies the Fourier identity that converts the bilinear average into an integral of linear averages of rotated functions.","marker":"[9]"},{"why":"Supplies the oscillation method and pointwise ergodic estimates for arithmetic sets used in the polynomial lemmas.","marker":"[7]"},{"why":"Provides the pointwise ergodic theorem for arithmetic sets on which the maximal estimates for polynomial averages depend.","marker":"[6]"},{"why":"Develops the circle-method estimates and return-time sequences invoked in the proof of the cornerstone oscillation lemma.","marker":"[8]"},{"why":"Extends the circle-method estimates to polynomials evaluated at primes, giving the prime-polynomial maximal and oscillation estimates.","marker":"[33]"},{"why":"Contains the discrete maximal inequality used to derive the maximal inequality for the shift on the integers.","marker":"[23]"},{"why":"Provides the transference principle for averages and the spectral fact about the shift used in the rotated estimates.","marker":"[4]"},{"why":"Gives the expository account of the oscillation and circle-method arguments underlying the lemma proofs.","marker":"[37]"}],"fun_headline_variants":["Polynomial and prime bilinear ergodic averages converge a.e.","Bourgain bilinear theorem simplified and extended to polynomials","Bilinear averages along polynomials and prime numbers converge","New proof: bilinear ergodic averages with polynomial steps converge","Simple proof extends bilinear ergodic theorem to polynomial steps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's bridge to the general case of two distinct polynomials $P$ and $Q$ is an assertion after (8.2): the oscillation inequalities remain true when the summands are multiplied by a phase $e^{iR(n)\\theta}$, uniformly over all phases $\\theta$. The paper gives no proof or reference for this rotated version, so everything beyond the symmetric case $Q=-P$ stands on that unproved estimate.","fun_headline_variants_meta":{"raw":{"variants":["Polynomial and prime bilinear ergodic averages converge a.e.","Bourgain bilinear theorem simplified and extended to polynomials","Bilinear averages along polynomials and prime numbers converge","New proof: bilinear ergodic averages with polynomial steps converge","Simple proof extends bilinear ergodic theorem to polynomial steps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1649,"prompt_tokens":1109,"completion_tokens":540,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":725,"completion_tokens_details":{"reasoning_tokens":459}},"tokens_in":725,"tokens_out":540,"duration_ms":5845,"temperature":1.0,"reasoning_tokens":459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:05:24.943326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose $R(n)=n^2$, $Q(n)=n$, and for growing $K$ compute, on finite subsets of $\\mathbb Z$, the quantity $\\sum_{k=1}^K \\|\\sup_{N_k\\le N\\le N_{k+1}, N\\in S_\\rho} |\\frac{1}{N}\\sum_{n\\le N} e^{in^2\\theta}g(x+n)-\\frac{1}{N_{k+1}}\\sum_{n\\le N_{k+1}} e^{in^2\\theta}g(x+n)|\\|_{\\ell^2(\\mathbb Z)}$. If there are sequences $g\\in\\ell^2(\\mathbb Z)$ and phases $\\theta$ for which this grows like $\\sqrt{K}\\|g\\|_2$ rather than $o(\\sqrt{K})\\|g\\|_2$, the uniform rotated inequality (8.3) fails, and with it the paper's route from the proved case $Q=-P$ to arbitrary polynomial pairs.","supporting_citations":[{"cited_title":"Bourgain, Double recurrence and almost sure convergence, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier identity that converts the bilinear average into an integral of linear averages of rotated functions."},{"cited_title":"Bourgain, An approach to pointwise ergodic theorems, Spring er L.N.M., Vol","cited_arxiv_id":null,"evidence_quote":"Supplies the oscillation method and pointwise ergodic estimates for arithmetic sets used in the polynomial lemmas."},{"cited_title":"Bourgain, On the pointwise ergodic theorem on Lp for arithmetic sets, Israel Journal of Mathematics, Vol","cited_arxiv_id":null,"evidence_quote":"Provides the pointwise ergodic theorem for arithmetic sets on which the maximal estimates for polynomial averages depend."},{"cited_title":"Bourgain, Pointwise ergodic theorems on arithmetic sets, with an appen- dix on return time sequences (jointly with H","cited_arxiv_id":null,"evidence_quote":"Develops the circle-method estimates and return-time sequences invoked in the proof of the cornerstone oscillation lemma."},{"cited_title":"Nair, On polynomials in primes and J","cited_arxiv_id":null,"evidence_quote":"Extends the circle-method estimates to polynomials evaluated at primes, giving the prime-polynomial maximal and oscillation estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the discrete maximal inequality used to derive the maximal inequality for the shift on the integers."},{"cited_title":"Bellow, Transference principles in ergodic theory, Harmonic An alysis and Partial Differential Equations, edited by Michael Christ, Carlos E","cited_arxiv_id":null,"evidence_quote":"Provides the transference principle for averages and the spectral fact about the shift used in the rotated estimates."},{"cited_title":"Thouvenot, La convergence presque sˆ ure des moyennes ergodiques suiv- ant certaines sous-suites d’entiers (d’apr` es Jean Bourgain)","cited_arxiv_id":null,"evidence_quote":"Gives the expository account of the oscillation and circle-method arguments underlying the lemma proofs."}],"review_version":1}