{"id":"a5a4101c-2fca-4ecd-9db1-f05f0e390ca6","arxiv_id":"2607.28091","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Polylog-dense subsets of [N] and of the primes contain nontrivial configurations x+b₁m,…,x+bₖm for almost every coefficient vector b in wide ranges of scales.","lead":"Dense subsets of integers (and of the primes) are shown to contain almost all random two-variable linear patterns once density is only polylogarithmic. The result strengthens classical Szemerédi-type theorems when the pattern coefficients themselves are allowed to vary.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claims are precisely the almost-all configuration statements of Theorems 1.1–1.2. The only genuine limitation is the shorter B-range on the prime side, which the authors flag and which the reader correctly identifies; it is a technological restriction, not an internal inconsistency. The quantitative GvN, U^{1+} degree lowering, and densification arguments appear free of gaps once the polynomial losses are absorbed into the choice of c_k. A full specialist line-by-line check was not performed here either, so confidence remains moderate, but nothing rises to a load-bearing objection that would alter the ACCEPT verdict.","tokens_in":49759,"tokens_out":471,"duration_ms":11435,"concrete_test":"Independently re-derive the terminal-form bound (3.15) of Lemma 3.4 from the iterated Cauchy–Schwarz of Lemma 3.3 and the folding identity for A(c,h), then feed the resulting averaged U^1 lower bound into Corollary 3.6; confirm that the exponent 2^{k-1} and the log(2/δ) power in Prop. 3.7 match. If they do, the subsequent degree-lowering and densification steps rest on a verified foundation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption note (B≤exp(w^{1/2}) forced by local Euler factors and lattice boundary estimates in Prop. 7.4 / Rem. 7.5) is accurate but already explicit in the paper and does not undermine the stated claims. Theorem 1.1 holds in the wide integer range with only polylog density; Theorem 1.2 correctly restricts to the shorter prime range that the sieve technology presently supports. The GvN (Prop. 3.7), degree-lowering (Lemmas 4.5–4.7), densification (Thm. 6.7) and density-increment (Lem. 8.4) chain is internally coherent, with polynomial losses tracked throughout. No hidden circularity, missing hypothesis, or quantitative break appears in the written argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that every subset A of [N] with density at least (log N)^{-c_k} contains a nontrivial configuration x+b_1 m,...,x+b_k m (m in [N/B]) for all but an O_k((log N)^{-c_k})-proportion of coefficient vectors b in ((B/2,B]\\cap Z)^k, whenever (log N)^{1/c_k}\\le B\\le N exp(-(log N)^{c'_k}). An analogous statement holds for relatively polylog-dense subsets of the primes, but only in the shorter range B\\le exp((log N)^{c_k}). The argument proceeds by a quantitative generalised von Neumann theorem giving U^k control of the averaged counting operator R_H (Prop. 3.7), degree lowering of the dual function first to U^2 and then to the U^{1+} norm (Lemmas 4.5–4.7), densification under a two-scale linear-forms condition (Thm. 6.7), verification of that condition for a truncated GPY majorant (Prop. 7.4), and a density-increment iteration (Lem. 8.4).","tokens_in":49924,"tokens_out":823,"duration_ms":16277,"significance":"The results give a polylogarithmic density threshold for almost all translation-invariant linear configurations in two variables, substantially stronger than the best known bounds for fixed configurations (Kelley–Meka/Bloom–Sisask for 3-APs, Green–Tao for 4-APs, Leng–Sah–Sawhney for longer APs). The same threshold is obtained relatively in the primes, albeit in a shorter coefficient range forced by the sieve. The technical contributions—a polynomial-loss GvN uniform in the coefficient scale B, degree lowering all the way to U^{1+}, and a carefully truncated two-scale GPY majorant—are of independent interest and are tracked with explicit polynomial dependencies throughout. The limitations of the prime range are stated honestly (Rem. 7.5).","major_comments":[],"minor_comments":[{"comment":"The constant c_k is used both as a density exponent and (with a different value) as a range exponent; a brief remark in the statements of Theorems 1.1–1.2 that the same symbol may stand for different positive constants depending only on k would avoid any momentary confusion.","section":"Theorems 1.1 and 1.2"},{"comment":"In the proof of Lemma 4.7 the smoothing parameter A=10 is fixed without comment; a parenthetical that any A>1 works and that 10 is chosen only for convenience would make the dependence clearer.","section":"Lemma 4.7"},{"comment":"Remark 7.5 already notes that an “almost-all forms” linear-forms condition might enlarge the prime range of B. A one-sentence forward reference in the introduction (after the statement of Theorem 1.2) would help the reader anticipate this limitation.","section":"Introduction / Remark 7.5"},{"comment":"Typographical: “Parithmetic progression” in (1.9) should be “P arithmetic progression”; a few other missing spaces appear in the same display.","section":"Equation (1.9)"}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the logical chain is complete and the quantitative losses are tracked carefully. I see no load-bearing gap. The shorter prime range is a genuine limitation of current sieve technology rather than an oversight; the authors flag it correctly. Suitable for a strong analytic-number-theory or additive-combinatorics journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is clean: for almost every coefficient vector b of size B in a wide range, polylog density already forces the configuration x+b_i m in [N], and the same holds for relatively polylog-dense subsets of the primes once B is restricted to exp((log N)^c). That is substantially better than the known worst-case thresholds for fixed long APs.\n\nWhat is new is the quantitative generalised von Neumann theorem that recovers polynomial U^{1+} control uniformly while the coefficients range over a growing box, the degree-lowering argument adapted to that extra averaging (concatenation to U^k, then dual-difference interchange + major-arc control down to U^{1+}), and the densification transfer that makes the same inverse theorem work for functions dominated by a two-scale GPY majorant. The architecture is coherent: iterated CS \to concatenation \to GvN, degree lowering, densification under an explicit linear-forms condition, majorant verification, density increment. Polynomial losses are tracked throughout; the citations to Peluse–Prendiville, Green–Tao and Shiu are used in the expected places.\n\nThe only real soft spot is the shorter B-range on the prime side. It is forced by local Euler-factor obstructions for large prime divisors of slope differences and by lattice-point boundary estimates in the GPY expansion; the authors flag it themselves in Remark 7.5 and note that an “almost-all forms” pseudorandomness assumption might enlarge the range. That is an acknowledged technological limit, not a hidden break, and it does not touch Theorem 1.1.\n\nThe math looks solid on a full read; no circularity, no fitted parameters fed back into the theorems, no citation oddities. This is for people who work on quantitative Szemerédi-type theorems or primes in linear configurations. It deserves a serious referee. I would accept it for peer review and would cite the integer result.","headline":"Solid quantitative additive-combinatorics paper: polylog density forces almost-all random translation-invariant configurations in [N], and relatively dense primes in a shorter range, via a new uniform GvN + degree-lowering to U^{1+}.","tokens_in":50648,"tokens_out":502,"would_cite":true,"duration_ms":12180,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11N13","37A45"],"pacs":[],"model":"grok-4.5","headline":"Polylog-dense sets of integers and primes contain almost every random linear configuration.","keywords":["linear configurations","Gowers norms","U^{1+} norm","degree lowering","generalised von Neumann","primes","densification","density increment"],"falsifier":"Exhibit a subset of [N] denser than N/(log N)^{c} that avoids the configuration x+b_i m for a positive-density set of b in ((B/2,B])^k inside the claimed range of B, or show that the truncated GPY majorant fails the linear-forms condition for some admissible system when B exceeds exp(w^{1/2}).","tokens_in":50549,"feed_emoji":"≈","tokens_out":1024,"duration_ms":16317,"temperature":0.7,"pith_summary":"The paper shows that once a subset of the first N integers is denser than a fixed power of 1/log N, it already contains nontrivial patterns of the shape x plus b_i times m for almost every choice of the coefficient vector b of size B, across a very wide range of B. The same conclusion holds for subsets of the primes that are only polylogarithmically dense relative to the primes, though the allowed range of B is shorter. The patterns are the natural translation-invariant linear configurations in two variables; arithmetic progressions are the special case b_i = i. Because the coefficients are allowed to grow, classical Gowers-norm control loses uniformity; the authors restore it by averaging over b, then lower the resulting high-degree control all the way to the U^{1+} norm, which detects long arithmetic progressions. A densification step transfers the bounded result to the primes via a truncated sieve majorant. The upshot is a quantitative existence theorem that is far stronger than what is known for any fixed large coefficient vector.","feed_headline":"Polylog-dense sets hold almost every random linear pattern","feed_subtitle":"Averaging over coefficients restores control that fixed large patterns lose, even inside the primes","key_machinery":"A quantitative generalised von Neumann theorem that controls the averaged counting operator R_H by the U^{1+} norm: after iterated Cauchy–Schwarz and concatenation produce U^k control, degree-lowering (dual-difference interchange plus major-arc analysis of phases) reduces the degree to U^{1+}, which is strong enough for a density-increment argument.","core_discovery":"Every subset of [N] denser than N/(log N)^{c_k} contains a nontrivial configuration x+b_1 m,...,x+b_k m with m in [N/B] for all but an O((log N)^{-c_k}) proportion of coefficient vectors b of size B, provided (log N)^{1/c_k} ≤ B ≤ N exp(-(log N)^{c'_k}). The identical statement holds for relatively polylog-dense subsets of the primes when B is at most exp((log N)^{c_k}).","pith_inferences":["If the open 'almost-all forms' pseudorandomness condition suggested in the paper can be verified, the prime result would reach the same coefficient range as the integer result.","The same averaging-plus-degree-lowering strategy should apply to other sparse or unbounded settings once a suitable two-scale majorant is available.","Quantitative bounds for almost-all configurations may be convertible into effective bounds for a positive-density set of explicit coefficient vectors by a second-moment argument."],"forward_implications":["Almost every translation-invariant linear pattern in two variables appears in every polylog-dense set of integers, far beyond the range known for any fixed pattern with large coefficients.","The same almost-everywhere statement holds inside the primes once relative density exceeds a power of 1/log N, for coefficients up to exp((log N)^c).","The U^{1+} inverse theorem and the averaged concatenation estimates become available as black-box tools for other random or averaged configuration problems.","Density-increment arguments that previously required fixed small coefficients now run uniformly for growing random coefficients."],"fun_headline_variants":["Polylog-dense sets contain almost all random linear configs","Dense [N] subsets hold nearly every x+b_i m pattern","Almost all coefficient vectors work in polylog-dense sets","Random linear patterns arise in dense sets and primes","Polylog density yields almost every random linear configuration"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The transfer to the primes needs a truncated sieve majorant that obeys a two-scale linear-forms condition only when the coefficient size B stays below exp of the square root of the small-prime level; larger B creates local Euler-factor obstructions the paper does not remove.","fun_headline_variants_meta":{"raw":{"variants":["Polylog-dense sets contain almost all random linear configs","Dense [N] subsets hold nearly every x+b_i m pattern","Almost all coefficient vectors work in polylog-dense sets","Random linear patterns arise in dense sets and primes","Polylog density yields almost every random linear configuration"]},"model":"grok-4.5","effort":"low","cost_usd":0.004395,"raw_usage":{"total_tokens":1255,"prompt_tokens":675,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":43948000,"prompt_tokens_details":{"text_tokens":675,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":516,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":675,"tokens_out":64,"duration_ms":10879,"temperature":1.0,"reasoning_tokens":516,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T18:03:52.507223+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a subset of [N] denser than N/(log N)^{c} that avoids the configuration x+b_i m for a positive-density set of b in ((B/2,B])^k inside the claimed range of B, or show that the truncated GPY majorant fails the linear-forms condition for some admissible system when B exceeds exp(w^{1/2}).","supporting_citations":[],"review_version":1}