{"id":"51ec41f3-6df0-4129-b2c2-ec3a28106e86","arxiv_id":"2412.14644","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A filtered exponential integrator is proved to converge for stochastic nonlinear wave equations with rough initial data, with rates up to O(τ^{2γ−}) in one and two dimensions.","lead":"This paper presents a time-stepping scheme for the stochastic nonlinear wave equation that keeps high accuracy when the initial data are rough or discontinuous, a regime where standard methods lose convergence. It proves error rates that improve on existing methods and includes experiments on discontinuous data in one and two dimensions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 is proved only for f≡0; the f≠0 case is asserted by analogy, but the scheme's deterministic term has no consistency analysis.","rationale":"The reader's weakest-assumption analysis correctly identifies the central gap: Theorem 2.1 is stated for general f, while the proof is carried out only for f≡0. My stress-test confirms that this is the most load-bearing issue rather than a minor presentational choice. The construction in Section 4 is genuinely tailored to the multiplicative noise term: the expansion of Σ(U(t_n+s)) about e^{sL}U(t_n), the cancellation Σ'Σ≡0 in (4.12), and the negative-norm estimates in Lemmas 3.5–3.7 all concern only the stochastic term. The deterministic term in (2.10) is never inserted into the error recursion, and no analogue of Lemma 4.1–4.3 is proved for F. Simply asserting that f≠0 can be handled 'in the similar way' is not sufficient, because the stochastic analysis relies on martingale properties and the algebraic structure of Σ, whereas F requires deterministic low-regularity estimates of a different kind. The numerical experiments in Section 5 all set f≡0 (they specify only σ(u)), so they provide no independent evidence for the f≠0 statement. I do not find a separate fatal flaw in the f≡0 analysis: the negative-norm estimates are plausible, the one-step consistency decomposition (4.14) is coherent, and the Gronwall argument in (4.32)–(4.35) is standard. However, those parts establish only the f≡0 version of Theorem 2.1. The appropriate verdict remains CONDITIONAL: the paper should either restrict Theorem 2.1 to f≡0 or supply a complete error analysis for the deterministic term. Because this is exactly the reader's stated condition, no verdict adjustment is needed.","tokens_in":21176,"tokens_out":19128,"duration_ms":140081,"concrete_test":"Set σ≡0 and f(u)=u (or f(u)=sin u), d=1, γ=1/4, with initial data whose Fourier coefficients decay like |k|^{-γ-ε}. For the scheme (2.10), measure the one-step deterministic error D(t_n)=U(t_n+τ)−[e^{τL}U(t_n)+τ e^{τL}Π_{τ^{-1}}F(Π_{τ^{-1}}U(t_n))] in the H^0×H^{-1} norm for τ=2^{-6},…,2^{-12}, using a highly resolved reference solution. If E||D(t_n)||_0^2 scales like τ^{1+4γ}=τ^2, the f≠0 case fits the existing Gronwall argument and Theorem 2.1 survives; if it scales only like τ^{1+2γ}=τ^{3/2} or worse, the claimed τ^{2γ} rate for nonzero f does not follow and the theorem must be restricted to f≡0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central convergence statement, Theorem 2.1, is stated for the full nonlinear drift f under condition (2.3), and the scheme (2.10) contains the term τ e^{τL} Π_{τ^{-1}} F(Π_{τ^{-1}} U^n). However, all of Section 4 analyzes only the multiplicative noise term. Lemma 4.1–4.3 bound R1, I2, and R2, all of which involve Σ and the Itô integral; the error recursion (4.32)–(4.35) contains no deterministic remainder coming from F. The paper explicitly says at the end of Section 2, 'we will focus on the case f(u) ≡ 0 in the rest of this paper' and later states the general case can be handled 'in the similar way.' This is not a routine extension: the construction for Σ exploits the special identity Σ'Σ ≡ 0 in (4.12) and the martingale structure of the noise, neither of which applies to F. For f≠0 there is an additional one-step error of the form ∫_0^τ e^{(τ-s)L} F(U(t_n+s)) ds − τ e^{τL} Π_{τ^{-1}} F(Π_{τ^{-1}} U(t_n)), and it is not shown that this term satisfies the same bound as (4.31). Indeed, known low-regularity integrators for the deterministic nonlinear wave equation require nontrivial resonance corrections to reach rates like 2γ for rough data. Thus the theorem as stated is not established; the proven result is the f≡0 case. This is exactly the gap identified by the reader, and it is load-bearing because every numerical experiment in Section 5 also takes f≡0.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a filtered low-regularity exponential integrator (2.10) for the stochastic nonlinear wave equation with multiplicative Itô noise and rough initial data in H^γ × H^{γ−1}. Theorem 2.1 claims mean-square convergence rates in L^2 × H^{−1}: O(τ^{2γ−}) in one and two dimensions for γ ∈ (0,1/2], O(τ^γ) in three dimensions for γ ∈ (0,1/2], and O(τ^{2γ−1/2−}) in three dimensions for γ ∈ (1/2,3/4]. The proof strategy combines the variation-of-constants formula, a Taylor expansion of the noise coefficient σ around the linear flow, frequency localization at the scale τ^{−1}, and negative-norm estimates for the resulting remainder terms. Numerical experiments in one and two dimensions with discontinuous and rough initial data compare the method with Euler–Maruyama and stochastic trigonometric integrators.","tokens_in":21465,"tokens_out":13468,"duration_ms":94042,"significance":"For the f ≡ 0 case, the paper contains a coherent derivation of the stated convergence rates, and the rates improve on existing methods under the same low regularity assumptions. The analysis is genuinely a priori: there are no fitted parameters, and the convergence rates are consequences of the stated estimates. If the theorem is corrected to cover exactly what is proved, namely the multiplicative-noise stochastic wave equation with f ≡ 0, this is a solid contribution and appears to be the first convergence proof in the regime below H^{1/2} × H^{−1/2}. However, the main theorem as stated claims convergence for the general nonlinearity f, while the proof and all numerical experiments treat only f ≡ 0; the significance of the paper is therefore conditional on either proving the nonlinear case or restricting the claim.","major_comments":[{"comment":"Theorem 2.1 is stated for a general nonlinearity f satisfying (2.3), and the scheme (2.10) contains the deterministic term τ e^{τL} Π_{τ^{-1}} F(Π_{τ^{-1}} U^n). However, the proof in Section 4 is carried out only for f ≡ 0. The text states this explicitly at the end of Section 2, and Lemmas 4.1–4.3 bound only the remainders R1, I2, and R2 arising from the multiplicative noise. The remainder in (4.30)–(4.31) and the error recursion in (4.32)–(4.35) contain no contribution from the drift F. The assertion that the general case can be handled “in the similar way” is not a proof: the one-step consistency error of the deterministic term, namely ∫_0^τ e^{(τ−s)L} F(U(t_n+s)) ds − τ e^{τL} Π_{τ^{-1}} F(Π_{τ^{-1}} U(t_n)), is not bounded at the required rate, and known low-regularity integrators for deterministic nonlinear wave equations need nontrivial resonance corrections to reach such rates. Since every numerical experiment in Section 5 also takes f ≡ 0, the theorem as stated is not established. The authors should either prove the f ≠ 0 case or state and prove the theorem for f ≡ 0, revising the abstract and introduction accordingly.","section":"Section 2 (Theorem 2.1) and Section 4"},{"comment":"The proof of Lemma 4.2 is terse at the point where the estimate E∥I2∥_1^2 ≲ τ^4 is derived. The argument needs a precise bound for ∥Σ′(e^{sL}U(t_n))(e^{(s−δ)L} − I)Σ(U(t_n+δ))∥_1. The displayed computation jumps from this norm to (s−δ)^2 times a bound involving Σ′LΣ. The intended bound can likely be justified using |sin(x)/x| ≤ 1 for the first component of (e^{hL} − I)Σ, but that justification is not given. Since this lemma is used in the final remainder estimate, the proof should be completed or the step should be stated as a separate estimate.","section":"Lemma 4.2"}],"minor_comments":[{"comment":"The abstract says the method achieves convergence for initial data in H^γ × H^{γ−1} “for all γ > 0,” while Theorem 2.1 states explicit rates only for γ ∈ (0,1/2] in d = 1,2 and γ ∈ (0,3/4] in d = 3. Please align the abstract with the theorem, or state what is proved for γ beyond these ranges.","section":"Abstract and Theorem 2.1"},{"comment":"Lemma 3.5, which supplies the one- and two-dimensional negative-norm estimates, is essential for the d = 1,2 rates in Lemma 4.3 but is imported from the authors’ paper [6] without proof. If [6] is not yet available, the present paper is not self-contained; please state these estimates as assumptions or reproduce their proofs.","section":"Lemma 3.5"},{"comment":"The notation τ^{2γ−} and τ^{4γ−} with a trailing “−” is informal. It would be clearer to say explicitly that the bounds hold for every ε > 0 with constants depending on ε, or to define the “−” convention once in Section 2.","section":"Notation throughout"},{"comment":"There are several typographical issues: “prseented” in Example 5.3, “walk-clock time” in Figures 2, 4, 6, and “L2(Ω) × H−1(Ω)” in captions where the spatial domain is O. These should be corrected.","section":"Section 5 and figure captions"},{"comment":"The notation in (5.1) writes U_N^{n+1} on both sides of the first line; the high-frequency recovery step in Algorithm 1 also writes U^{T/τ} where U_N^{T/τ} is meant. Please clarify the notation for the fully discrete variable.","section":"Algorithm 1 and (5.1)"}],"recommendation":"major_revision","confidential_remarks":"The f ≡ 0 result is a genuine contribution and the core estimates appear sound, so this is not a reject. The main problem is that the paper's headline theorem overclaims: it is stated for a general nonlinear drift f, but the analysis and all experiments cover only f ≡ 0. I would require the authors to either supply the missing deterministic error analysis or explicitly restrict the theorem and the paper's claims to the multiplicative-noise equation. If they restrict the claim, the paper will be publishable after a moderate revision; if they keep the general claim, the missing analysis is substantial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the f=0 result: a filtered exponential integrator with a rigorous mean-square convergence proof for multiplicative noise and initial data below H^{1/2} × H^{-1/2}. The rates in Theorem 2.1 for that case are supported by a coherent analysis—the Taylor expansion of Σ, the cancellation Σ'Σ ≡ 0, and the new 3D negative-norm estimates (Lemmas 3.4, 3.6) are real work. If the f=0 case is what the paper claims, it is the first proof in that regime and worth publishing.\n\nThe soft spot is also exactly where the reader and stress-test put it. Theorem 2.1 is stated for a general nonlinear drift f under condition (2.3), but every proof in Section 4 is for f ≡ 0, and the authors say the general case can be handled \"in the similar way.\" That is not a routine extension. The analysis of the stochastic term leans on the martingale structure and the special identity Σ'Σ = 0, which does not apply to the deterministic term F. The extra one-step error from e^{τL}Π_{τ^{-1}}F(Π_{τ^{-1}}U^n) is never bounded, and known low-regularity integrators for the deterministic nonlinear wave equation need nontrivial resonance corrections. So the theorem as stated is not established; the proven, and numerically tested, result is the f=0 case. The paper should either restrict Theorem 2.1 to f=0 or supply the missing deterministic analysis.\n\nSmaller issues: the 1D/2D negative-norm composition estimates are cited from the authors' prior work [6] without proof—acceptable if that paper is available, but a referee should check it. The numerical experiments are consistent with the proven rates but have no error bars, no shared code, and reference solutions come from the same method; that modestly weakens the empirical confirmation but is not disqualifying for a numerical analysis paper.\n\nBottom line: the mathematical core for f=0 is solid, the novelty is clear, and the paper deserves a serious referee. I would send it to review with a request to fix the scope of the main theorem and, if the general f claim is kept, to provide the actual estimates. My own verdict would be conditional acceptance on those terms.","headline":"New convergence rates for the stochastic wave equation with rough data are real for f=0, but Theorem 2.1 overreaches: the general nonlinearity is asserted, not proved.","tokens_in":22056,"tokens_out":1313,"would_cite":true,"duration_ms":10524,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M12","65M15","35Q55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs a filtered exponential integrator that computes rough solutions of the stochastic nonlinear wave equation with initial data in $H^{\\gamma} \\times H^{\\gamma-1}$, proving mean-square rates up to $\\tau^{2\\gamma-}$ in one…","keywords":["stochastic nonlinear wave equation","low regularity","high order convergence","error estimates","exponential integrator","multiplicative noise","rough initial data","Fourier spectral method"],"falsifier":"Run the scheme (2.10) on the one-dimensional equation with a nonzero smooth drift, say $f(u)=\\sin(u)$, noise $\\sigma(u)=16\\sin(u)$, and rough initial data in $H^{1/4}\\times H^{-3/4}$, and estimate the Monte Carlo mean-square $L^2\\times H^{-1}$ error over decreasing step sizes $\\tau$; if the observed convergence slope falls clearly below the claimed $\\tau^{1/2-}$ as $\\tau\\to 0$, the unproved $f\\neq 0$ extension is false.","tokens_in":20916,"feed_emoji":"🌊","tokens_out":11024,"duration_ms":83711,"temperature":0.7,"pith_summary":"This paper claims that a filtered exponential integrator can compute rough solutions of the stochastic nonlinear wave equation with multiplicative Itô noise at convergence rates that were previously out of reach. For initial data in $H^{\\gamma} \\times H^{\\gamma-1}$, the scheme is proved to achieve mean-square error of order $\\tau^{2\\gamma-}$ in one and two dimensions for $\\gamma \\in (0,\\tfrac12]$ and $\\tau^{\\max(\\gamma, 2\\gamma-\\tfrac12-)}$ in three dimensions for $\\gamma \\in (0,\\tfrac34]$. These are the first proven convergence rates for rough stochastic wave solutions below the regularity threshold $H^{1/2} \\times H^{-1/2}$, where classical methods lose order and can develop spurious oscillations. The proof is carried out for the case $f(u)\\equiv 0$, with the authors stating that the general nonlinear drift can be handled similarly.","feed_headline":"New integrator doubles known rate for rough stochastic waves","feed_subtitle":"Mean-square errors fall like $\\tau^{2\\gamma}$ in 1D and 2D, the first proof below $H^{1/2}\\times H^{-1/2}$.","key_machinery":"The machinery is the filtered low-regularity exponential integrator (2.10), which evolves the linear wave semigroup $e^{\\tau L}$ and projects the data and nonlinearities through the frequency-localization operator $\\Pi_{\\tau^{-1}}$. Three estimates carry the proof: the identity $\\frac{d}{ds} e^{-sL}\\Sigma(e^{sL}U) = e^{-sL}(-\\sigma(\\tilde u), \\sigma'(\\tilde u)\\tilde v)^\\top$, used to control how the noise coefficient changes along the linear flow; negative-norm bounds such as $\\|\\sigma'(\\Pi_N u)\\Pi_N v\\|_{H^{-1}} \\lesssim N^{1-2\\gamma+}$ in one and two dimensions and $N^{\\tfrac32-2\\gamma+}$ in three dimensions; and a frequency-localization estimate for $\\Sigma(U)-\\Sigma(\\Pi_N U)$ in $L^2\\times H^{-1}$. A second-order Taylor expansion of $\\Sigma$ around $e^{sL}U(t_n)$ splits the stochastic increment into a dominant term $\\Pi_{\\tau^{-1}}\\Sigma(\\Pi_{\\tau^{-1}}U^n)\\Delta_n W$ and remainders of size $\\tau^3$, $\\tau^4$, and $\\tau^{1+4\\gamma-}$ (or $\\tau^{4\\gamma-}$ in 3D), which are then summed through a discrete Gronwall argument.","core_discovery":"The central claim is Theorem 2.1: for the semilinear stochastic wave equation $\\partial_{tt}u - \\Delta u = f(u) + \\sigma(u)\\,dW$ on the $d$-dimensional torus, the filtered low-regularity exponential integrator $U^{n+1} = e^{\\tau L}U^n + \\tau e^{\\tau L}\\Pi_{\\tau^{-1}}F(\\Pi_{\\tau^{-1}}U^n) + e^{\\tau L}\\Pi_{\\tau^{-1}}\\Sigma(\\Pi_{\\tau^{-1}}U^n)\\Delta_n W$ has mean-square $L^2 \\times H^{-1}$ error of order $\\tau^{2\\gamma-}$ in one and two dimensions and $\\tau^{\\max(\\gamma, 2\\gamma-\\tfrac12-)}$ in three dimensions whenever the initial pair lies in $H^{\\gamma}\\times H^{\\gamma-1}$. The discovery is that one can avoid Hölder continuity of the exact solution in time entirely: the noise coefficient is expanded in a Taylor series along the linear wave flow, the error terms $R_1$, $I_2$, and $R_2$ are bounded using Itô isometry and negative-norm estimates for $\\sigma'(\\Pi_N u)\\Pi_N v$, and the term $\\tau e^{\\tau L}\\Pi_{\\tau^{-1}}F(\\Pi_{\\tau^{-1}}U^n)$ is included for the drift. The resulting rates double the previously known order in one and two dimensions for rough data, and give the first proof of convergence below $H^{1/2}\\times H^{-1/2}$; the proof as written restricts the error analysis to $f\\equiv 0$.","pith_inferences":["Beyond the paper, the frequency-filtering mechanism suggests a template for other stochastic dispersive equations: project the SPDE data and noise coefficient at the time-step scale, evolve everything above that scale exactly with the linear flow, and Taylor-expand the nonlinearity along that flow.","The dimension-dependent negative-norm estimates imply the practical gain is largest in one and two dimensions; in three dimensions, for very rough data with small $\\gamma$, the proven rate drops to $\\tau^{\\gamma}$, so the benefit over classical methods is smaller and should not be oversold.","A direct test of the unproved $f\\neq 0$ extension is to run (2.10) with a nonzero smooth drift and compare empirical mean-square rates with Theorem 2.1; the paper's own experiments focus on the multiplicative noise case, so the general-drift claim remains the main open check."],"forward_implications":["In one and two dimensions, rough initial data in $H^{\\gamma}\\times H^{\\gamma-1}$ with $\\gamma\\in(0,\\tfrac12]$ are computed at rate $\\tau^{2\\gamma-}$, twice the previously available rate under the same regularity.","In three dimensions the scheme is proved to converge at rate $\\tau^{\\gamma}$ for $\\gamma\\in(0,\\tfrac12]$ and $\\tau^{2\\gamma-\\tfrac12-}$ for $\\gamma\\in(\\tfrac12,\\tfrac34]$, extending proven convergence below $H^{1/2}\\times H^{-1/2}$.","The fully discrete version with Fourier spectral discretization and high-frequency recovery costs $O(N^d\\log(N)^d T/\\tau + N^{\\alpha d})$ overall, because the high-frequency part is recovered once as $e^{TL}\\Pi_{(N,N^\\alpha]}U^0$ instead of being stepped every time level.","Piecewise smooth and discontinuous initial data can be evolved without the spurious oscillations seen with semi-implicit Euler-Maruyama and stochastic trigonometric schemes in the numerical experiments.","If the asserted extension to $f\\neq 0$ holds, the same scheme applies to semilinear stochastic wave equations with smooth bounded nonlinearities, not only to the pure noise case analyzed in the proof."],"supporting_citations":[{"why":"Supplies the well-posedness result and the variation-of-constant formula (2.7) for the stochastic wave equation with rough initial data, which is the starting point of the error analysis.","marker":"[28]"},{"why":"Supplies the one- and two-dimensional negative-norm estimates for composition functions used in Lemma 3.5 and the frequency-localization estimate used in Lemma 3.7; it is also the source of the low-regularity integrator ideas.","marker":"[6]"},{"why":"Provides the identity (3.6) for the derivative of the noise coefficient along the linear wave flow, which is the core structural identity behind the scheme's construction.","marker":"[19]"},{"why":"Establishes the first strong convergence result (order 1/2) for the 1D stochastic wave equation and serves as the classical baseline whose regularity requirements are relaxed.","marker":"[26]"},{"why":"Achieves strong convergence of order one for multiplicative noise with Verlet-type integrators, one of the baselines the new method improves on in the rough-data regime.","marker":"[4]"},{"why":"Develops a 1.5-order scheme for the stochastic nonlinear wave equation under $H^1\\times L^2$ regularity, illustrating the order reduction that motivates the low-regularity construction.","marker":"[13]"},{"why":"Supplies the Bernstein inequalities stated in Lemma 3.3, which control the error introduced by low- and high-frequency projections throughout the proof.","marker":"[14]"}],"fun_headline_variants":["Integrator doubles rate for rough stochastic wave data","First convergence proof below H^{1/2} for stochastic waves","Exponential integrator hits tau^{2gamma} error for rough waves","Filtered scheme doubles known rate for rough wave equations","Rough stochastic wave solver achieves improved error rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stated rates are proved only when the equation's nonlinear drift term is zero; the theorem as stated for a general drift depends on the authors' assertion, made without carrying out the analysis, that the same error bounds follow, and on earlier cited estimates that are used without proof here.","fun_headline_variants_meta":{"raw":{"variants":["Integrator doubles rate for rough stochastic wave data","First convergence proof below H^{1/2} for stochastic waves","Exponential integrator hits tau^{2gamma} error for rough waves","Filtered scheme doubles known rate for rough wave equations","Rough stochastic wave solver achieves improved error rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00072,"raw_usage":{"total_tokens":3359,"prompt_tokens":1200,"completion_tokens":2159,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":816,"completion_tokens_details":{"reasoning_tokens":2078}},"tokens_in":816,"tokens_out":2159,"duration_ms":18730,"temperature":1.0,"reasoning_tokens":2078,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:03:11.268873+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the scheme (2.10) on the one-dimensional equation with a nonzero smooth drift, say $f(u)=\\sin(u)$, noise $\\sigma(u)=16\\sin(u)$, and rough initial data in $H^{1/4}\\times H^{-3/4}$, and estimate the Monte Carlo mean-square $L^2\\times H^{-1}$ error over decreasing step sizes $\\tau$; if the observed convergence slope falls clearly below the claimed $\\tau^{1/2-}$ as $\\tau\\to 0$, the unproved $f\\neq 0$ extension is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the well-posedness result and the variation-of-constant formula (2.7) for the stochastic wave equation with rough initial data, which is the starting point of the error analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the one- and two-dimensional negative-norm estimates for composition functions used in Lemma 3.5 and the frequency-localization estimate used in Lemma 3.7; it is also the source of the low-regularity integrator ideas."},{"cited_title":"To appear in ESAIM:M2AN","cited_arxiv_id":null,"evidence_quote":"Provides the identity (3.6) for the derivative of the noise coefficient along the linear wave flow, which is the core structural identity behind the scheme's construction."},{"cited_title":"Walsh: On numerical solutions of the stochastic wave equation","cited_arxiv_id":null,"evidence_quote":"Establishes the first strong convergence result (order 1/2) for the 1D stochastic wave equation and serves as the classical baseline whose regularity requirements are relaxed."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Achieves strong convergence of order one for multiplicative noise with Verlet-type integrators, one of the baselines the new method improves on in the rough-data regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops a 1.5-order scheme for the stochastic nonlinear wave equation under $H^1\\times L^2$ regularity, illustrating the order reduction that motivates the low-regularity construction."},{"cited_title":"Guo: Spectral Methods and Their Applications","cited_arxiv_id":null,"evidence_quote":"Supplies the Bernstein inequalities stated in Lemma 3.3, which control the error introduced by low- and high-frequency projections throughout the proof."}],"review_version":1}