{"id":"979b2c2a-5ccf-4727-b345-05e67127f970","arxiv_id":"2610.03417","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The shifted primes P-1 contain infinitely many sum-product patterns {x, x+y, xy} and longer {x, x+y, ..., x+ky, xy}, a result containing Green-Tao as a special case.","lead":"Two number theorists show that the set of primes minus one, P-1, contains infinitely many patterns {x, x+y, xy} with x and y arbitrarily large distinct integers, and longer variants {x, x+y, ..., x+ky, xy} for any fixed k. The statement contains the Green-Tao theorem as a special case and extends the primes-pattern program to patterns that mix addition with multiplication.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The proof hinges on unpublished companion preprints; Theorem 3.2 and [5, Props 4.5/6.1] must be verified against exactly the hypotheses used here.","rationale":"The reader's weakest_assumption identifies exactly the two unpublished external dependencies: Theorem 3.2 from [8] and Propositions 4.5 and 6.1 from [5]. My reading of the present paper confirms that these are genuinely load-bearing: Proposition 2.3 (hence Step 1) collapses without Theorem 3.2, and Proposition 2.10(iii) (hence the final lower bound) explicitly invokes [5] with no in-text proof. The paper's internal structure is coherent and I found no internal contradiction, parameter fitting, or circular reasoning. The concern is purely about verifiability of external estimates. I agree with the reader's assessment that the verdict should be UNVERDICTED, and my stress-test does not change that. The concrete test is to audit the companion papers, which is the only way to settle whether the dependency holds. I also checked that the application ranges in the text (H ~ N^{3/4}, X ~ N, w2 = exp((log N)^{1/10})) satisfy the stated hypotheses of Theorem 3.2 as quoted, so the issue is not a mismatch within the present paper but the external validity of the quoted bound.","tokens_in":36839,"tokens_out":2450,"duration_ms":16336,"concrete_test":"Obtain the manuscript [8] and verify that its Theorem (the local Gowers bound) is stated with precisely the same normalization of the interval Gowers norm, the same Siegel model Λ_tilde_W (including the n^{β−1} factor and the treatment of the trivial character when no Siegel zero exists), and, critically, that its hypotheses H ≥ X^{5/8+ε} and w ≤ exp((log X)^{1/10}) hold for the values used in Proposition 2.3: X = ∆D/d b_{j0} x + 1 with x ∈ [N^{9/10}, N] and H = 4∆² D M, M = N^{3/4}. Independently, cross-check [5, Prop 4.5 and Prop 6.1] against the application in Section 7: the polynomial P(y) = y(∆dy+1) must be squarefree and have no fixed prime divisor, the sieve weight f must satisfy the stated decay condition |f(p,y)−1| ≤ 1_{p|P(y)} + C/p with C=6, and the range w0 ≤ exp((log M)^{1/2}) must be respected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central input is Theorem 3.2, quoted from [8] (Teräväinen, 'Preprint'), asserting a local Gowers norm bound ||Λ−Λ_tilde_W||_{U^s(X,X+H]} ≪ w^{-a_s} + exp(−(log X)^γ) for H ≥ X^{5/8+ε}. This is used in Proposition 2.3 to control every non-zero Gowers error term SG,t, and any weakening or narrowing of its hypotheses (e.g. requiring H ≥ X^{1−ε} or w much smaller) would invalidate the bound exp(−(log N)^{2δ}) on which the first main estimate rests. Similarly, the final positivity step (proof of Proposition 2.10(iii)) invokes [5, Prop 6.1 and Prop 4.5] from a separate arXiv preprint (2405.12190) with overlapping authorship. The paper itself contains no independent check of these results and does not state the precise hypotheses of [5] beyond what is needed in the application, so the chain of deductions leading to Theorem 1.6 is only as secure as two unpublished sources. This is not a circularity or internal inconsistency; it is a load-bearing dependency on external claims that the present text does not make verifiable. If either result fails or has hypotheses that do not align exactly with the ranges used (X ~ N, H ~ N^{3/4}, w = exp((log N)^{1/10}) for Theorem 3.2; the specific polynomial P(y)=y(∆dy+1) and the weight f(p,y) for [5]), then the main theorem loses its proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper proves that the shifted primes P−1 = {p−1} contain infinitely many configurations {x, xy, L_1(x,y), …, L_k(x,y)} for arbitrary pairwise non-proportional linear forms L_j(x,y) = a_j x + b_j y (Theorem 1.3), in particular {x, x+y, …, x+ky, xy} for any k (Corollary 1.4), a statement that subsumes the Green–Tao theorem. The engine is the weighted asymptotic Theorem 1.6, assembled in four steps: (1) replacement of the von Mangoldt factors at the linear forms by a Siegel model, using a local Gowers-norm bound (Propositions 2.2–2.3); (2) replacement of the outer Λ(ψ0(Dmn)) factor via a Bombieri–Vinogradov estimate with an additional large coprime modulus and Siegel correction (Propositions 2.4–2.5); (3) passage to a singular series by sieve and Weil bound arguments (Propositions 2.6–2.8); (4) evaluation of the positive main term (Propositions 2.9–2.10). Most technical machinery (Lemmas 3.1, 3.3, 5.1, 6.1, 6.2 and the claims in Section 7) is proved in the paper; the two exceptions are the central Gowers-uniformity input, Theorem 3.2, quoted from the unpublished companion paper [8], and the final positivity input [5, Propositions 4.5 and 6.1] from an arXiv preprint.","tokens_in":37222,"tokens_out":12960,"duration_ms":78236,"significance":"If correct, the result is a genuinely new structural statement about shifted primes: arbitrarily long arithmetic progressions together with a product element xy, i.e., mixed additive–multiplicative structure, with Green–Tao as a special case. The paper's own contributions are substantial and I verified them internally: the 4-step assembly is arithmetically consistent (the exponent bookkeeping in Section 4, the reduction to a square in Proposition 2.3, the interval-length check X^{11/16} ≤ H ≤ X, the parameter ranges in Sections 5–7 all work out), Lemma 3.3 is a real two-variable Bombieri–Vinogradov variant proved in detail, and Lemma 6.1 (fundamental-lemma asymptotic for the short-interval average) and Lemma 6.2 (Weil-bound character sum) are complete. Lemma 3.1, Lemma 5.1 and the sieve arguments are also proved in text. There is no parameter fitting, and the claim (1.5) is a falsifiable asymptotic. The assessment therefore hinges on the two load-bearing external results, neither of which is verifiable from the present text.","major_comments":[{"comment":"Theorem 3.2 is the sole input controlling every non-trivial Gowers error term S_{G,t} in the first main estimate, yet it is quoted from [8], cited only as a 'Preprint', with no proof and no statement of hypotheses beyond the four lines of the theorem itself. In its application to (4.4) the parameters are X ∈ [N^{9/10}, N^{1+o(1)}], H = 4Δ²DM ≈ N^{3/4+o(1)}, s = 2k+3, w = w2 = exp(1/2 (log N)^{1/10}), and the conclusion must beat exp(−(log N)^{2δ}) after multiplication by p² ≈ (DM)². The reader cannot verify from the present text (i) that the interval-length hypothesis H ≥ X^{5/8+ε} is satisfied for a specific chosen ε (any ε with (5/8+ε)·9/10 ≤ 3/4 works, but ε is never fixed in Section 4); (ii) that the exponent γ = γ(2k+3, ε) of the exp(−(log X)^γ) term satisfies γ ≥ 2δ, which the final bound exp(−(log N)^{2δ}) requires; and (iii) that the quoted bound applies to the Siegel-corrected function Λ − Λ̃_W rather than to Λ alone, and to intervals of exactly this shape. Please either prove this bound in the present paper or reproduce the companion theorem's full hypotheses and constants verbatim and exhibit the parameter alignment; as it stands, Propositions 2.2–2.3 and hence the first main estimate rest on an unverifiable claim.","section":"Section 3.2, Theorem 3.2; used in Section 4 (proof of Proposition 2.3)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The lower bound (2.4), which is the only source of the positive constant in Theorem 1.6, is delivered by two applications of results from the arXiv preprint [5]: Proposition 4.5 supplies the Siegel-model error over moduli Δdr ≤ w0 in Claim 1, and Proposition 6.1 is applied at scale M with P(y) = y(Δdy+1), C = 6, and w0 = exp((log N)^{1/3}) ≤ exp((log M)^{1/2}). Neither proposition is stated, and the verification of their hypotheses is compressed into a single sentence ('The other hypotheses of that proposition hold with C = 6 for sufficiently large N, since the coefficients of P are M^{o(1)} …'), which invokes the condition |f(p,y)−1| ≤ 1_{p|P(y)} + 6/p without reproducing the actual hypothesis. The reader therefore cannot check that M = N^{3/4}, w1 = exp((log N)^{1/4}), and w0 = exp((log N)^{1/3}) lie inside the admissible parameter set of [5]. Please state [5, Propositions 4.5 and 6.1] with their full hypotheses and confirm the alignment of ranges, or give self-contained proofs.","section":"Section 7, proof of Proposition 2.10(iii) (Claims 1 and 2 and the appeal that follows)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The letter D is used both for the exponent D = ⌊(log N)^δ⌋ and for the product D = p_1^D ··· p_ℓ^D; equations such as (1.5) and (1.10) are hard to parse on first reading. Please rename one of them (for example, write the exponent as e or κ).","section":"Section 1.1, equation (1.4)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'The theorem gives, for some γ = γ(k) > 0' misstates the dependence: in Theorem 3.2 the constant is γ = γ(s, ε). Also, the chosen value of ε for the application is never fixed; please state ε (e.g., ε = 1/16) and record the check H ≥ X^{5/8+ε} with that value, together with the resulting γ.","section":"Section 4, proof of Proposition 2.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The paper switches between the notations eβ, eχ (mod e q) for Siegel objects at levels w, w1, w2, w0. The level is usually clear from context, but a one-line table listing (w_i, eβ_i, eχ_i, e q_i) and the convention for their absence would reduce the risk of confusion in Sections 5–7, where eβ_0, e q_0 first appear without a Definition 1.9-style entry.","section":"Section 1.2 and Definition 2.1","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The two load-bearing external inputs are both unpublished: [8] is a companion preprint by one of the authors (Teräväinen), and [5] (arXiv:2405.12190) has overlapping authorship. This is not a circularity or an internal contradiction — the four-step assembly is coherent and the in-paper lemmas I checked are sound — but it means the manuscript's main theorem is exactly as secure as two papers in preprint form whose hypotheses are not even stated. I would ask the editor to require, as a condition of acceptance, either (a) that the authors reproduce the full statements of [8, Theorem] and [5, Propositions 4.5 and 6.1] with an explicit parameter-alignment check for the ranges used here, or (b) that the companion proofs be supplied as an appendix. The paper is otherwise in good shape for this journal: the internal machinery (Lemmas 3.1, 3.3, 5.1, 6.1, 6.2, Proposition 2.10(i)–(ii)) is novel and proved in detail, and the main claim, if verified, is a substantial advance over Green–Tao for shifted primes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The news here is real: P−1 carries arbitrarily long progressions together with a product term — {x, x+y, ..., x+ky, xy}, and more generally {x, xy, L_1(x,y), ..., L_k(x,y)} for arbitrary non-proportional linear forms. That is a new class of configuration; the Green–Tao and linear-equations-in-primes program never folded in the multiplicative factor xy. The claim that Green–Tao is a special case is correct. I also appreciate the honest scoping in Proposition 1.2: they state plainly that {x, y, xy} patterns would force a solution to a hard Liouville problem, so they don’t oversell the limitations of the method.\n\nWhat the paper does well: the architecture is coherent and most of it is genuinely proved in-paper. Lemma 3.1 (L1 type-I approximation), Lemma 3.3 (Bombieri–Vinogradov with Siegel correction and an extra congruence), Lemma 5.1, Lemma 6.1, and the Weil-bound work in Lemma 6.2 are all developed here in detail, and the four main-term propositions (2.2, 2.4, 2.6, 2.9) assemble cleanly into Theorem 1.6. The deduction of Theorem 1.3 from Theorem 1.6 through the d-m-n expansion is standard and checks out. No circularity, no parameter fitting.\n\nThe soft spot is real and it is exactly where the reader put it. Theorem 3.2, the local Gowers norm bound for Λ − Λ̃_W over intervals of length just above X^{5/8+ε}, is quoted from companion preprint [8] by co-author Teräväinen, and every non-zero Gowers error term S_{G,t} passes through it. The final positivity step leans on [5, Props 6.1 and 4.5] from a preprint with overlapping authorship. The authors do not hide this: they state Theorem 3.2 is “established in a companion-paper.” I checked the parameter ranges at the point of invocation: X ~ N, H = 4Δ²DM ≥ X^{11/16}, and the theorem is used with H ≥ X^{5/8+ε}. Since 11/16 sits ε ≤ 1/16 above 5/8, comfortably inside the stated ε < 3/16, the invocation is sound if the theorem is true as stated. The issue is verification, not architecture.\n\nWho this is for: everyone in the primes/transference program. It deserves a serious referee, and the referee’s job is to read [8] and [5] and confirm their hypotheses match the ranges used here exactly. My recommendation: engage, with the companion-paper dependency as the condition for acceptance.","headline":"A genuinely new mixed-additive-multiplicative result in the shifted primes, well-architected and mostly self-contained, but load-bearing on two unpublished companion preprints a referee must verify.","tokens_in":37886,"tokens_out":2902,"would_cite":true,"duration_ms":16702,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11N05","11B30","11N36","11N37"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"The paper claims $\\mathbb{P}-1$ contains $\\{x,x+y,xy\\}$ for arbitrarily large distinct $x,y$, and $\\{x,x+y,\\ldots,x+ky,xy\\}$ for any fixed $k$, a statement containing the Green–Tao theorem as a special case.","keywords":["shifted primes","sum-product patterns","Green–Tao theorem","Siegel zeros","Gowers norms","von Mangoldt function","Bombieri–Vinogradov theorem","additive combinatorics"],"falsifier":"Verify Theorem 3.2 directly: for $k=1$ (so $s=5$), $X = 10^{12}$, and $w_2 = \\exp(\\tfrac12(\\log X)^{1/10})$, compute $\\|\\Lambda - \\tilde{\\Lambda}_{W_2}\\|_{U^5(X,X+H]}$ on a window of length $H = X^{3/4}$; exceeding $w_2^{-a_5} + \\exp(-(\\log X)^\\gamma)$ would falsify the bound that powers Step 1. A lighter check: scan $x,y \\le 10^{10}$ and test whether $\\{x, x+y, xy\\} \\subset \\mathbb{P}-1$ occurs at the density the lower bound (1.5) predicts — systematic absence far below that density would contradict Theorem 1.1.","tokens_in":38198,"feed_emoji":"🔢","tokens_out":10573,"duration_ms":57067,"temperature":0.7,"pith_summary":"The paper aims to prove that the shifted primes — the numbers $p-1$ for primes $p$ — contain patterns that mix addition and multiplication: for arbitrarily large distinct $x,y$, each of $x$, $x+y$, and $xy$ is a shifted prime. The stronger form claims the same for $\\{x, x+y, \\ldots, x+ky, xy\\}$ for any fixed $k$, a statement whose progression part alone reproduces the Green–Tao theorem. A reader should care because it exhibits, for the first time, genuinely joint additive–multiplicative structure inside a sparse prime-related set, not just one or the other. The proof is a four-step approximation that swaps the von Mangoldt function for a Siegel-corrected model at two scales and leans on a local Gowers norm estimate imported from a companion preprint.","feed_headline":"Shifted primes host sum-product patterns of any length","feed_subtitle":"The set p−1 contains {x, x+y, …, x+ky, xy} for every k, with Green–Tao as a special case.","key_machinery":"The engine is the Siegel model $\\tilde{\\Lambda}_W(n) = \\Lambda_W(n)(1 - \\tilde{\\chi}(n)n^{\\tilde{\\beta}-1})$, the truncated von Mangoldt function at level $W = \\prod_{p \\le w} p$ corrected by a factor built from the Siegel zero $\\tilde{\\beta}$ and its character $\\tilde{\\chi}$ at level $w$ (taken as $\\tilde{\\beta}=1$, $\\tilde{\\chi}\\equiv 0$ when no zero exists). The proof has four steps: replace the $k$ inner factors of $\\Lambda$ by $\\tilde{\\Lambda}_{W_2}$ (Step 1, on the local Gowers norm bound of Theorem 3.2, quoted from a companion preprint); replace the outer factor by the $\\tilde{\\Lambda}_{W_1}$ model via a Bombieri–Vinogradov estimate with an extra congruence (Step 2, Lemma 3.3); remove the Siegel corrections and pass to a singular series of local densities using sieve and Weil bounds (Step 3); and evaluate the positive main term (Step 4, using a positivity estimate quoted from another preprint). The dilating primes $p_1, \\ldots, p_\\ell$ are chosen to avoid the Siegel character moduli $Q$, which is what produces the $\\tau(D)$ saving in the lower bound.","core_discovery":"The central claim is Theorem 1.6: a weighted count of triples $(d,m,n)$, evaluating the von Mangoldt function at $k+2$ affine forms, is $\\gg \\tau(D)\\varphi(W_1)MN/W_1$ for all large $N$. Since each factor forces a prime value, this lower bound produces many $d,m,n$ with $\\Delta dm + 1$, $\\Delta^2Dmn + 1$, and $L_j(\\Delta dm, \\Delta D n/d) + 1$ all prime, and re-labelling $x = \\Delta dm$, $y = \\Delta D n/d$ yields the patterns $\\{x, xy, L_1(x,y), \\ldots, L_k(x,y)\\}$ inside $\\mathbb{P}-1$. From here the paper derives Theorem 1.3 (arbitrary non-proportional linear forms $L_j = a_j x + b_j y$) and Corollary 1.4 ($\\{x, x+y, \\ldots, x+ky, xy\\}$). The authors also state the boundary of their result: finding $\\{x,y,xy\\}$, let alone $\\{x,y,x+y,xy\\}$, in $\\mathbb{P}-1$ for arbitrarily large $x,y$ would settle a major open problem on the Liouville function, so the pattern's form is essentially the strongest possible.","pith_inferences":["The four-step approximation is largely modular: any sparse set whose prime-detecting function satisfies the same Gowers and Bombieri–Vinogradov inputs should inherit the same mixed patterns, so the genuinely new ingredient is the product-plus-progression combination rather than this particular set.","The interval-length exponent $5/8$ in the imported Gowers bound sets the effective scale of the proof; improving that exponent would likely sharpen the main-term savings or widen the admissible range of the forms, though the paper does not pursue this.","A quantitative version for $\\mathbb{P}-a$ with general fixed $a$ seems within reach of the same machinery, but the paper itself only treats $\\mathbb{P}-1$.","Because the Liouville obstruction (Proposition 1.2) blocks any pattern containing both $x$ and $y$ themselves, $\\{x, x+y, xy\\}$ is plausibly the strongest mixed pattern current methods can extract, and any advance toward $\\{x, y, xy\\}$ must bypass that obstruction."],"forward_implications":["$\\mathbb{P}-1$ contains infinitely many triples $\\{x, x+y, xy\\}$ with $x,y$ arbitrarily large distinct — this is Theorem 1.1.","For every fixed $k$, $\\mathbb{P}-1$ contains $\\{x, x+y, \\ldots, x+ky, xy\\}$; dropping the product element leaves the Green–Tao progressions, which Corollary 1.4 contains as a special case.","The pattern persists for any family of non-proportional linear forms $L_j(x,y) = a_j x + b_j y$ (Theorem 1.3).","The weighted count in (1.5) is bounded below by a positive multiple of $\\tau(D)\\varphi(W_1)MN/W_1$, so the pattern is not merely present but dense in the natural counting sense."],"supporting_citations":[{"why":"Supplies the local Gowers norm bound (Theorem 3.2) on intervals of length $\\ge X^{5/8+\\varepsilon}$; the Step 1 error estimate (Proposition 2.2) rests on it.","marker":"[8]"},{"why":"Cited for its Propositions 4.5 and 6.1, used in Proposition 2.10(iii) to obtain the final positivity lower bound (2.4).","marker":"[5]"},{"why":"Handbook source used throughout for the fundamental lemma of sieves, Weil bounds on character sums, the Landau–Page theorem, and the large sieve.","marker":"[4]"},{"why":"Supplies the generalised von Neumann theorem (Proposition 4.1) that bounds multicorrelations along linear forms by Gowers norms.","marker":"[3]"},{"why":"The Green–Tao theorem, the special case that Corollary 1.4 contains and the benchmark this result extends.","marker":"[2]"},{"why":"Shiu's Bombieri–Vinogradov-type bound for multiplicative functions, used in Lemma 3.1 and in the type II estimates of Lemma 3.3.","marker":"[7]"}],"fun_headline_variants":["Shifted primes contain sum-product patterns of any length","Prime-minus-one set hosts {x, x+y, …, x+ky, xy} for all k","Sum-product patterns of arbitrary length found in shifted primes","Shifted primes yield sum-product patterns, Green–Tao as special case","New theorem: shifted primes contain sum-product patterns for every k"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing input is the local Gowers norm bound of Theorem 3.2, that $\\Lambda - \\tilde{\\Lambda}_W$ has small $U^s$-norm on every interval of length at least $X^{5/8+\\varepsilon}$, which the paper quotes from a companion preprint without proof here; the first main-term estimate (Proposition 2.2) collapses if that bound is false or holds only under narrower hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Shifted primes contain sum-product patterns of any length","Prime-minus-one set hosts {x, x+y, …, x+ky, xy} for all k","Sum-product patterns of arbitrary length found in shifted primes","Shifted primes yield sum-product patterns, Green–Tao as special case","New theorem: shifted primes contain sum-product patterns for every k"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"cost":0.000198128,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000198128,"upstream_inference_prompt_cost":0.000100128,"upstream_inference_completions_cost":9.8e-05},"total_tokens":1270,"prompt_tokens":920,"completion_tokens":350,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":255}},"tokens_in":920,"tokens_out":350,"cost_source":"openrouter_usage","duration_ms":3205,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":255,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-05T07:34:12.528325+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Verify Theorem 3.2 directly: for $k=1$ (so $s=5$), $X = 10^{12}$, and $w_2 = \\exp(\\tfrac12(\\log X)^{1/10})$, compute $\\|\\Lambda - \\tilde{\\Lambda}_{W_2}\\|_{U^5(X,X+H]}$ on a window of length $H = X^{3/4}$; exceeding $w_2^{-a_5} + \\exp(-(\\log X)^\\gamma)$ would falsify the bound that powers Step 1. A lighter check: scan $x,y \\le 10^{10}$ and test whether $\\{x, x+y, xy\\} \\subset \\mathbb{P}-1$ occurs at the density the lower bound (1.5) predicts — systematic absence far below that density would contradict Theorem 1.1.","supporting_citations":[{"cited_title":"and Host, B","cited_arxiv_id":null,"evidence_quote":"Supplies the local Gowers norm bound (Theorem 3.2) on intervals of length $\\ge X^{5/8+\\varepsilon}$; the Step 1 error estimate (Proposition 2.2) rests on it."},{"cited_title":", TITLE =","cited_arxiv_id":null,"evidence_quote":"Cited for its Propositions 4.5 and 6.1, used in Proposition 2.10(iii) to obtain the final positivity lower bound (2.4)."},{"cited_title":"arXiv e-prints , keywords =","cited_arxiv_id":null,"evidence_quote":"Handbook source used throughout for the fundamental lemma of sieves, Weil bounds on character sums, the Landau–Page theorem, and the large sieve."},{"cited_title":"and Zheng, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the generalised von Neumann theorem (Proposition 4.1) that bounds multicorrelations along linear forms by Gowers norms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Green–Tao theorem, the special case that Corollary 1.4 contains and the benchmark this result extends."},{"cited_title":"and Host, B","cited_arxiv_id":null,"evidence_quote":"Shiu's Bombieri–Vinogradov-type bound for multiplicative functions, used in Lemma 3.1 and in the type II estimates of Lemma 3.3."}],"review_version":1}