{"id":"a3a9f296-b97b-42fd-826d-4f0786ef4bf6","arxiv_id":"2502.01818","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Deterministic local wellposedness on R x T at s>3/4 (or s>1/2 under a low-frequency condition), shown optimal for the bilinear/Picard method, plus probabilistic wellposedness for generic H^s data with s>-1/26.","lead":"This paper lowers the local wellposedness threshold for the Zakharov-Kuznetsov equation on a cylinder to s > 3/4 in general, and to s > 1/2 for data satisfying a natural low-frequency condition. It also proves these thresholds are optimal for Picard iteration, and obtains almost sure solutions below L2 for generic randomized data.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the X^{s,b} s>3/4 result appears to fail in the bad-interaction region: the bound for I_2 in Section 2.5.2 gives N^{1/2} I_2 / (L1 L2 L3)^{1/2} of order N^{1/4} for L_i ~ 1, not O(N^{\\varepsilon/2}), so estimate (2.40) is not established.","rationale":"I read the paper in good faith and found a substantial, concrete issue that the reader's verdict did not flag. The reader identified Corollary 2.10 (geometric rigidity) as the weakest assumption. I agree that corollary is load-bearing, and its proof has a harmless sign typo (the rotation sign in the displayed bound for |(\\zeta,m)-(-k/2,-k/2)|), but the conclusion still holds up to rearrangement, so I do not see a fatal flaw there. The more serious problem is in the power counting of the bad-interaction estimate for the X^{s,b} result. The paper reduces the X^{s,b} Sbad estimate to (2.40), but the bound it proves for I_2 is too weak by a factor N^{1/4} in the worst case L_1=L_2=L_3=1. The specific counterexample used later for optimality (Theorem 1.5) has exactly those parameters and saturates the loss, so the failure of (2.40) is not merely a loose constant but a genuine defect in the presented argument. This does not prove Theorem 1.3 false; the final estimate (2.6) might still hold, but the reduction through (2.40) is invalid as written. Consequently, the deterministic X^{s,b} claim (s>3/4) is not supported by the current proof, and the paper should be accepted only conditionally on repairing this estimate (for example, by extracting a second factor from the volume of B_\\Theta or by a different decomposition of the resonant region). The Y^{s,b} and probabilistic parts appear to rely on the i=1 bound, which is not affected, so they may remain valid.","tokens_in":59546,"tokens_out":61778,"duration_ms":490710,"concrete_test":"Take the explicit functions from the proof of Theorem 1.5 with R_1=R_2=[-N^{-1/2},N^{-1/2}], k=N, m=n=-N/2, and L_1=L_2=L_3=1; normalize so that \\|u\\|_{L^2}=\\|v\\|_{L^2}=\\|w\\|_{L^2}=1. Compute the reduced quantity I_2 from (2.41) and then N^{1/2} I_2/(L_1 L_2 L_3)^{1/2}. If this is \\sim N^{1/4} for large N (rather than O(N^{\\varepsilon/2})), then estimate (2.40) is false and the proof of Theorem 1.3 must be revised. Independently, re-derive (2.47) keeping a second factor from the measure of B_\\Theta in the (\\nu,\\xi) variables to check whether the correct power is \\Theta^{1/2} rather than \\Theta^{1/4}.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The X^{s,b} part of Theorem 1.3 (s>3/4) depends on the Sbad estimate (2.6), which is reduced in Remark 2.2 to the dyadic sufficient condition (2.11)/(2.39) and then to (2.40) for I_i. For i=2 (the X case), Section 2.5.2 derives the bound I_2 \\lesssim L_med^{1/2} L_max^{1/4} (\\log N)/N^{1/4}, displayed after (2.47). Substituting this into the left side of (2.40) gives N^{1/2} I_2 / (L1 L2 L3)^{1/2} \\approx N^{1/4} (\\log N) / (L_max^{1/4} L_min^{1/2}), which is at least N^{1/4} \\log N when L_max = L_min = 1. This contradicts (2.40), which requires the left side to be at most N^{\\varepsilon/2} for arbitrarily small \\varepsilon. The paper states that '(2.40) is satisfied for I_2 as long as s>3/4', but the displayed bound does not support this: it is off by a power N^{1/4}. The resonant example used in the proof of Theorem 1.5 (k=N, m=n=-N/2, |\\nu|,|\\omega| \\lesssim N^{-1/2}, and L_1=L_2=L_3=1) saturates the estimate, so the loss is real and not an artifact of crude constants. Because (2.40) is the only mechanism controlling the Sbad contribution to the X^{s,b} bilinear estimate, this leaves a gap in the proof of Theorem 1.3. The Y^{s,b} part (i=1) has the additional |\\nu|^{1/2} factor and does not suffer this loss.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Zakharov–Kuznetsov equation on the cylinder R×T. It proves local well-posedness in X^{s,b} for every s>3/4 and in Y^{s,b} for every s>1/2 under the natural low-frequency condition, and it shows that these regularity ranges are optimal for the corresponding bilinear estimates up to endpoints. The method is a dyadic decomposition of the resonant set, based on a geometric rigidity result saying that near-resonant frequencies must lie close to a specific triangular configuration. The paper also proves almost sure local well-posedness for generic randomized initial data in H^{s1} for every s1>-1/26, with the remainder lying in Y^{s,b}, s>1/2, using a novel Z^{s,b} norm that combines an L2-based Y^{s,b} part with an L8-based auxiliary part.","tokens_in":59975,"tokens_out":24953,"duration_ms":225301,"significance":"If the results are correct, this is a substantial improvement over the previous deterministic well-posedness range in R×T and provides the first sub-L2 almost sure result in this geometry, with a smoothing remainder. The deterministic part is written in considerable detail, with explicit dyadic estimates, a geometric classification of the resonant set, and concrete counterexamples for the sharpness statements. The paper also introduces an L8-based auxiliary norm in the context of random-data well-posedness; this is a promising idea that may be applicable elsewhere. The main external input, Proposition 2.3 from Molinet–Pilod, is properly cited and used as a black box. No parameter fitting or circular reasoning is apparent.","major_comments":[{"comment":"The displayed moment computation Eω(g_k g_m g_{k'}g_{m'}) is not correct for the complex Gaussians defined in (1.10). For standard complex Gaussians, Isserlis/Wick pairing pairs holomorphic factors with antiholomorphic factors, so the correct expression in the second integral of (3.12) should involve Eω(g_k g_m \\overline{g_{k'}}\\overline{g_{m'}}). With complex Gaussians, the nondegenerate expectation Eω(g_k g_m g_{k'}g_{m'}) is zero for the three pairings listed, and the third pairing (k,k')=-(m,m') does not arise for the holomorphic term (Sptq u_ω^0)^2; it arises only for the mixed term Sptq u_ω^0 \\overline{Sptq u_ω^0}. The subsequent case analysis of Proposition 3.2 is therefore not justified as written. The proof can likely be repaired by using real Gaussians throughout the probabilistic part, or by carrying the antiholomorphic factors through and checking that the two genuine pairings for the holomorphic term satisfy the same estimates, but this is a load-bearing correction.","section":"§3.1, Eq. (3.12)"},{"comment":"The quantification of ε in the reduction leading to (2.40) is not explicit, and the sentence after (2.47) saying that '(2.40) is satisfied for I2 as long as s>3/4' does not follow from the displayed bound without further explanation. The dyadic reduction in (2.7) fixes ε with δ<<ε<<s-1/2, while (2.40) asks for a bound by N^{ε/2} with no explanation of which ε is meant. Substituting the bound I2 ≤ L_med^{1/2} L_max^{1/4} (log N)/N^{1/4} gives N^{1/2}I2/(L1L2L3)^{1/2} ≈ N^{1/4} (log N)/(L_min^{1/2}L_max^{1/4}), which exceeds N^{ε/2} for small ε, in particular for ε<s-1/2<1/2 when s<1. To make the argument valid, the authors should state that the ε in (2.40) is an auxiliary exponent that may be taken larger than 1/2, and that the final dyadic ε in (2.39) is then chosen just above 1/4+9δ, which is admissible whenever s>3/4. With this clarification the alleged N^{1/4} gap disappears; as written, however, the proof of (2.6) is incomplete.","section":"§2.5, Eqs. (2.40) and (2.47)"}],"minor_comments":[{"comment":"A similar ε-quantification point applies to the harmless logarithmic loss in the bound for I1: log N is bounded by N^{ε/2} for any fixed ε>0, but this should be stated explicitly, since the text currently claims (2.40) without discussing the ε convention.","section":"§2.5.1, after Eq. (2.45)"},{"comment":"In the definition of M-interactions and in the sentence after (2.30), the notation 'mint|ν|, |ζ|, |ξ|u' is a typesetting artifact; the intended expression is min{|ν|,|ζ|,|ξ|}. Please correct all such instances.","section":"§2.3.2, Definition 2.8"},{"comment":"The argument that N3=Nmin in subcase A would benefit from one more sentence: after using N3≤6 min{N1,N2} and Nmax≥8Nmin, the contradiction with N3=Nmax should be written out explicitly, since the current text is terse.","section":"§2.2.3, Subcase A"},{"comment":"Remark 3.4 treats the case (k,k')=-(m,m') as part of the holomorphic term; after correcting the moment formula, this case should be moved to the analysis of the mixed term Sptq u_ω^0 \\overline{Sptq u_ω^0}, or the authors should explain why overcounting this case is harmless for the upper bound.","section":"§3.1, Remark 3.4"},{"comment":"Several inline displays are corrupted by OCR artifacts, for example 'N 1{2' for N^{1/2} and 'L 1{2} i' for L_i^{1/2}. The authors should carefully proofread the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The deterministic part appears essentially correct, and the suspected N^{1/4} gap in the bad-interaction estimate is, in my view, resolvable by an explicit ε-hierarchy statement rather than by a substantive change. The probabilistic part, however, contains an incorrect Gaussian moment formula in the proof of Proposition 3.2; this is a load-bearing point that must be corrected. The paper is a strong contribution and the issues seem fixable, but the revision should be sent back for a careful check of the moment computations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the stress-test note is right. In Section 2.5.2, substituting the displayed bound for I_2 into (2.40) gives N^{1/4} log N / (L_min^{1/2} L_max^{1/4}) when L_min = L_max ~ 1, not N^{ε/2}. The example used in Theorem 1.5 has exactly L_1 = L_2 = L_3 = 1, so the loss is not an artifact of crude constants; it is a real failure of the written estimate. Because (2.40) is the only mechanism controlling the S_bad contribution to the X^{s,b} bilinear estimate, Theorem 1.3 is not established as written. The paper even says \"(2.40) is satisfied for I_2 as long as s>3/4,\" but (2.40) contains no s, which is a warning sign that something was mis-stated.\n\nWhat is genuinely new and good here: the geometric rigidity result identifying the triangular resonant configuration is a real insight, and the S_good/S_bad decomposition built on it is clean and useful. The Y^{s,b} result at s>1/2 appears to go through, since the extra |ν|^{1/2} factor in (2.5) rescues the analogous estimate; I could not find a similar loss in Section 3. The Theorem 1.5 counterexamples are concrete and convincing. The probabilistic part is also a serious contribution: the L8-based auxiliary norm Z^{s,b} is a natural tool, and the first below-L2 well-posedness on R x T is a meaningful step toward Gibbs measure constructions.\n\nSoft spots, in proportion: the gap in Section 2.5.2 is load-bearing for one of the two deterministic theorems, so the advertised s>3/4 result should not be accepted on the strength of this version. Everything else—the Y^{s,b} theorem, the optimality examples, and the probabilistic framework—deserves serious consideration. The paper is honest with its external inputs: Molinet-Pilod's Proposition 2.3 is prior work, and there is no circular reasoning or self-citation. I did not machine-verify every dyadic inequality, and the case analysis is long enough that a careful referee should check Section 2 in full.\n\nWho is this for? People working on low-regularity well-posedness for ZK-type equations, resonant set analysis, and randomized initial data. The Z^{s,b} norm idea is likely to be reused elsewhere.\n\nRecommendation: send it to peer review, but expect a major revision on Section 2.5.2. The Y^{s,b} half and the probabilistic half are worth refereeing even if the X^{s,b} half needs repair.","headline":"A genuinely promising paper with a real load-bearing gap in the s>3/4 X^{s,b} proof; the Y^{s,b} and probabilistic parts look solid enough to justify referee time.","tokens_in":60531,"tokens_out":4975,"would_cite":false,"duration_ms":44127,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q53","35A01","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-resonant interactions in the ZK equation on a cylinder are forced into a triangle, and that geometry yields optimal local well-posedness and almost-sure solutions below L2.","keywords":["Zakharov–Kuznetsov equation","cylinder R x T","local well-posedness","resonant set","bilinear estimates","X^{s,b} spaces","Y^{s,b} spaces","randomized initial data"],"falsifier":"Solve the polynomial system $|\\Delta|\\leq N^2 M$, $|\\theta_i-\\theta_j|\\leq M$, $\\nu+\\zeta+\\xi=0$, $k+m+n=0$, and $N\\leq |(\\nu,k)|,|(\\zeta,m)|,|(\\xi,n)|\\leq 8N$, then measure the maximal distance of each triple to the triangle with vertices $(0,k)$, $(-k/2,-k/2)$, $(k/2,-k/2)$ over all rearrangements. If for a sequence with $M/N\\to 0$ this distance is not $O(M)$, the rigidity classification is false and the deterministic proof collapses; equivalently, an explicit sequence showing the bilinear estimate (1.7) diverges for some $s$ just above $1/2$ in $Y^{s,b}$ would refute the weighted well-posedness theorem.","tokens_in":59320,"feed_emoji":"📐","tokens_out":12794,"duration_ms":119096,"temperature":0.7,"pith_summary":"The paper's target is the local Cauchy problem for the Zakharov–Kuznetsov equation $u_t + \\partial_{x_1}\\Delta u = \\frac12\\partial_{x_1}(u^2)$ on the cylinder $\\mathbb{R}\\times\\mathbb{T}$, where the $x_2$-direction is periodic and non-dispersive. It establishes local well-posedness in Fourier restriction spaces for every regularity $s>3/4$ in $X^{s,b}$, and for every $s>1/2$ in the weighted space $Y^{s,b}$ when the data satisfy a natural low-frequency condition, and it proves these thresholds are optimal for the bilinear fixed-point estimates. The engine is a geometric description of the resonant set: whenever the phase sum $\\Delta$ and its directional derivatives are small, the three spatial frequencies must lie within an $O(M)$ neighborhood of a triangle with vertices $(0,k)$, $(-k/2,-k/2)$, $(k/2,-k/2)$, up to rearrangement. The paper also randomizes generic initial data and proves almost-sure local solutions for every $s_1 > -1/26$, with the nonlinear remainder in the smoother space $Y^{s,b}$, $s>1/2$. A reader should care because this is the first route below $L^2$ regularities on the cylinder and because the resonant-set geometry looks like a transferable structural fact about the equation.","feed_headline":"Triangle rigidity unlocks ZK equation at s > 1/2","feed_subtitle":"Near-resonant triples form a triangle, unlocking optimal well-posedness on R×T and almost-sure solutions below L2.","key_machinery":"The load-bearing object is the rigidity classification: if three comparable frequencies satisfy the first dyadic smallness condition, then up to rearrangement $|\\nu|\\lesssim M$, $|(\\zeta,m)-(-k/2,-k/2)|\\lesssim M$ and $|(\\xi,n)-(k/2,-k/2)|\\lesssim M$; this follows from a lemma saying that three zero-sum vectors of nearly equal length must be rotations of one another by $2\\pi/3$. Around that classification the proof builds a dyadic decomposition into $M$-interactions of first and second kind, uses two new bilinear estimates on boxes of side $\\sim M$ whose gains are $M\\sqrt{\\min(L_1,L_2)}$ and $1/(N^{1/2}\\sqrt{L_1L_2})$, and adds, for random data, the composite norm $Z^{s,b}$ consisting of $Y^{s,b}$ plus an $L^8$-in-frequency, $L^2$-in-time term that controls concentration of mass on small frequency sets.","core_discovery":"On the paper's own terms, the central discovery is that the resonant set of the Zakharov–Kuznetsov equation on $\\mathbb{R}\\times\\mathbb{T}$ is not a wild algebraic variety but a rigid, finite-type configuration. Concretely, for three frequencies $\\nu+\\zeta+\\xi=0$, $k+m+n=0$ of comparable size, the condition that $\\Delta=\\varphi(\\nu,k)+\\varphi(\\zeta,m)+\\varphi(\\xi,n)$ and its first directional derivatives are all small forces the three vectors to lie, up to rearrangement and $O(M)$ error, inside boxes around $(0,k)$, $(-k/2,-k/2)$, $(k/2,-k/2)$. The proof partitions the comparable-frequency regime into dyadic $M$-interactions, treats 'first-kind' interactions where $\\Delta$ is large by oscillation, 'second-kind' ones by refined bilinear estimates where the $\\theta_i$-differences are large, and confines the bad triangle region to a small set where the weight $x\\xi/|\\xi|$ in $Y^{s,b}$ (or a dedicated $L^8$-frequency norm in $Z^{s,b}$) supplies exactly the missing power. This yields the $s>1/2$ weighted and $s>3/4$ unweighted bilinear estimates, and the counterexamples in the paper's Theorem 1.5 show neither survives below those exponents, up to endpoints.","pith_inferences":["The paper leaves implicit that the same triangle rigidity should survive small perturbations of the dispersion relation, for instance anisotropic or higher-order variants of the Zakharov–Kuznetsov equation; if it does, the dyadic good/bad architecture would transfer directly.","Since the paper itself doubts optimality of $s_1>-1/26$, a natural next step is to re-run the $Z^{s,b}$ estimates with sharper high-low randomization lemmas; the explicit obstruction is the near-vertical low-frequency configuration where the $L^8$ control is only barely sufficient.","A concrete testable extension would be to replace Gaussians by any independent random variables with exponentially decaying tails in the randomization; the paper's argument uses only independence and tail decay, so verifying the theorem for, say, Bernoulli random variables would confirm the mechanism's robustness.","The endpoint cases are left open by the optimality statement: deciding whether $s=1/2$ in $Y^{s,b}$ or $s=3/4$ in $X^{s,b}$ actually fails would sharpen the boundary between the Picard method and whatever non-Picard argument might reach the critical regularity $s_c=-1$."],"forward_implications":["Deterministically, local well-posedness on $\\mathbb{R}\\times\\mathbb{T}$ now holds for all $s>3/4$ in $X^{s,b}$, and for all $s>1/2$ in $Y^{s,b}$ with the natural low-frequency condition, improving the previously known $s\\ge 1$ and $s>9/10$ results.","The $s>3/4$ and $s>1/2$ thresholds are optimal for the bilinear estimates or Picard iteration: below them the data-to-solution map cannot be $C^2$, so any improvement must use a genuinely non-Picard argument.","For generic randomized initial data, almost-sure local solutions exist in $H^{s_1}$ for every $s_1>-1/26$, with the remainder smoother than the linear evolution; this is the paper's first step below $L^2$ toward a Gibbs measure.","The $Z^{s,b}$ construction isolates a mechanism in which $L^\\infty$-in-frequency control beats $L^2$ control in very resonant regions, a mechanism the paper argues should apply to other dispersive problems where small sets of frequencies carry the obstruction."],"supporting_citations":[{"why":"Supplies the bilinear estimates used for the high-low and non-resonant cases; the paper's new estimates refine them.","marker":"[35]"},{"why":"Provides the standard $X^{s,b}$ theory showing that the bilinear estimates (1.7) imply local well-posedness via the Duhamel fixed-point argument.","marker":"[45]"},{"why":"Gives the prior two-dimensional well-posedness result and the model for the Picard-iteration failure argument used in the optimality theorem.","marker":"[24]"},{"why":"Introduces the $X^{s,b}$ spaces and the fixed-point framework on which the functional setup is built.","marker":"[2]"},{"why":"Initiates the random-data subtract-the-linear-evolution method that the probabilistic part adapts, though without the $TT^*$ trick.","marker":"[4]"},{"why":"Provides the frequency-block randomization with independent coefficients used to define generic initial data and the almost-sure regularity.","marker":"[34]"},{"why":"Provides precedent for $L^p$-based variants of $X^{s,b}$ that the auxiliary $L^8$-in-frequency term of $Z^{s,b}$ builds on.","marker":"[37]"}],"fun_headline_variants":["Resonant triangles straighten ZK well-posedness","Rigid resonances push ZK solvability to s>1/2","Triangle geometry cracks ZK equation's Cauchy threshold","Optimal ZK well-posedness from finite-type resonances","Resonant set rigidity beats bilinear estimate barrier"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the rigidity classification: that near-resonant, small-derivative frequency triples of comparable size are within a uniform constant multiple of $M$ of the triangle shape, and if those constants deteriorated with the dyadic scale, the case-4 estimates that produce $s>1/2$ and $s>3/4$ would not close.","fun_headline_variants_meta":{"raw":{"variants":["Resonant triangles straighten ZK well-posedness","Rigid resonances push ZK solvability to s>1/2","Triangle geometry cracks ZK equation's Cauchy threshold","Optimal ZK well-posedness from finite-type resonances","Resonant set rigidity beats bilinear estimate barrier"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":3063,"prompt_tokens":984,"completion_tokens":2079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":1992}},"tokens_in":600,"tokens_out":2079,"duration_ms":13910,"temperature":1.0,"reasoning_tokens":1992,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:21:25.237672+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the polynomial system $|\\Delta|\\leq N^2 M$, $|\\theta_i-\\theta_j|\\leq M$, $\\nu+\\zeta+\\xi=0$, $k+m+n=0$, and $N\\leq |(\\nu,k)|,|(\\zeta,m)|,|(\\xi,n)|\\leq 8N$, then measure the maximal distance of each triple to the triangle with vertices $(0,k)$, $(-k/2,-k/2)$, $(k/2,-k/2)$ over all rearrangements. If for a sequence with $M/N\\to 0$ this distance is not $O(M)$, the rigidity classification is false and the deterministic proof collapses; equivalently, an explicit sequence showing the bilinear estimate (1.7) diverges for some $s$ just above $1/2$ in $Y^{s,b}$ would refute the weighted well-posedness theorem.","supporting_citations":[{"cited_title":"Bilinear Strichartz estimate s for the Zakharov–Kuznetsov equation and applications","cited_arxiv_id":null,"evidence_quote":"Supplies the bilinear estimates used for the high-low and non-resonant cases; the paper's new estimates refine them."},{"cited_title":"Nonlinear Dispersive Equations","cited_arxiv_id":null,"evidence_quote":"Provides the standard $X^{s,b}$ theory showing that the bilinear estimates (1.7) imply local well-posedness via the Duhamel fixed-point argument."},{"cited_title":"Global well-posedness for the Cauchy pr oblem of the Zakharov–Kuznetsov equation in 2D","cited_arxiv_id":null,"evidence_quote":"Gives the prior two-dimensional well-posedness result and the model for the Picard-iteration failure argument used in the optimality theorem."},{"cited_title":"Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations: Part I: Schr¨ odinger equations","cited_arxiv_id":null,"evidence_quote":"Introduces the $X^{s,b}$ spaces and the fixed-point framework on which the functional setup is built."},{"cited_title":"Invariant measures for the 2D-defocusing nonlinear Schr¨ odinger equation","cited_arxiv_id":null,"evidence_quote":"Initiates the random-data subtract-the-linear-evolution method that the probabilistic part adapts, though without the $TT^*$ trick."},{"cited_title":"Random Data Cauchy Theo ry for Nonlinear Wave Equations of Power-Type on R3","cited_arxiv_id":null,"evidence_quote":"Provides the frequency-block randomization with independent coefficients used to define generic initial data and the almost-sure regularity."},{"cited_title":"In variant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS","cited_arxiv_id":null,"evidence_quote":"Provides precedent for $L^p$-based variants of $X^{s,b}$ that the auxiliary $L^8$-in-frequency term of $Z^{s,b}$ builds on."}],"review_version":1}