{"id":"6eba99ca-fe48-4b09-a907-245acc2caea0","arxiv_id":"2510.18024","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fixed large K, every subset of the y=log^K N smooth numbers up to N with positive relative density contains a nontrivial 3-term arithmetic progression.","lead":"This paper proves that any subset of the 'super smooth' numbers—integers up to N whose prime factors are all at most (log N)^K—that keeps a positive fraction of all such numbers must contain three equally spaced numbers. It extends a 2016 result of Harper by allowing the exponent K to stay fixed while the density threshold δ becomes arbitrarily small.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof runs transference on the composite modulus N_b and counts wrap-around modular 3-APs; the final step converting these to integer APs in A_b is invalid without a prime modulus >3N_b.","rationale":"I read the paper in good faith. The strategy—W-trick, transference, Harper restriction—is appropriate for the claimed theorem, and several apparent defects are repairable: Proposition 4.2's a=0 statement is a normalization slip (the normalized h has mean 1), and Proposition 4.3's use of Harper on A_b can be fixed by writing n=(m+b_2)/W with m y-smooth, so the coefficients lie on the smooth set. However, the central counting step in §4 is missing a necessary argument: the transference principle is applied to Z/N_bZ although the cited proposition requires a prime modulus, and the modular APs counted can wrap, so the asserted conversion to an integer AP is invalid. The concrete example shows the inference is not a harmless technicality. This is exactly the kind of gap that makes a proof CONDITIONAL rather than complete. Because the gap appears repairable (choose P>3N_b and transfer the estimates), I do not move the verdict to REJECT; I keep CONDITIONAL. The reader identified the prime-modulus aspect but did not emphasize the wrap-around conversion; hence partial agreement.","tokens_in":9055,"tokens_out":33138,"duration_ms":254448,"concrete_test":"To see the wrap-around step cannot be rescued as written, take N_b=5 and A_b={1,3,4}. Then (n,d)=(4,2) in Z/5Z has 4, 4+2=6≡1, 4+4=8≡3 all in A_b, yet the integers 4,1,3 are not an arithmetic progression. This directly refutes the claim that every nontrivial modular 3-AP in A_b yields an integer 3-AP in A_b. A full settlement of the proof's validity would be to re-run the argument with a prime P>3N_b and verify that the Fourier estimates of Propositions 4.2 and 4.3 hold on Z/PZ for the W-tricked functions; the above example shows the current modulus choice is insufficient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final paragraph of §4 applies Proposition 3.6 on Z/N_bZ and then converts a lower bound for sum_{n,d in Z/N_bZ} 1_{A_b}(n)1_{A_b}(n+d)1_{A_b}(n+2d) into the existence of an integer 3-AP in A_b. This is the central step of Theorem 1.1. It has two unaddressed problems. First, Proposition 3.6 explicitly assumes N is a large prime, but N_b = floor(N/(b_1W))+1 is generally composite. No composite-modulus analogue or reduction to a prime modulus is supplied; the Fourier/major-arc analysis leading to the needed lower bound is not obviously valid on Z/N_bZ. Second, even granting a composite-modulus transference, the count on Z/N_bZ is a count of modular APs, and these can wrap. Since A_b is a subset of {1,...,N_b} and the modulus equals the support bound, a modular AP such as residues 4,1,3 in Z/5Z is counted but is not an integer AP. The proof's concluding sentence, 'if the difference of this progression in A_b is d, then the arithmetic progression corresponds to another arithmetic progression ... with difference b_1Wd', requires the progression to be an integer progression, which has not been established. A repair is available by choosing a prime P>3N_b and working on Z/PZ with support in [1,P/3], but the paper does not carry this out; in particular, Propositions 4.2-4.3 are stated only for modulus N_b. This is the load-bearing gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims Theorem 1.1: for a fixed large K, y = log^K N, any δ>0 and all sufficiently large N, every subset A of the y-smooth numbers S(N,y) with |A| ≥ δ Ψ(N,y) contains a nontrivial 3-term arithmetic progression. The proof uses the W-trick, defines functions ν_b and f_b = 1_{A_b}ν_b on a set A_b (the W-tricked lift of A), states three supporting propositions (a weighted lower bound, Fourier decay, and an L^p restriction bound), and then applies the Green–Tao transference principle (Proposition 3.6) to obtain a lower bound for triple correlations of f_b. This is converted to a lower bound for triple correlations of 1_{A_b}, and the author concludes that a non-constant arithmetic progression exists in A_b and hence, after multiplying the common difference by b_1 W, in A.","tokens_in":9424,"tokens_out":9449,"duration_ms":81677,"significance":"If correct, the result would be a genuine extension of Harper's theorem: it fixes the parameter K rather than allowing K to grow as δ tends to 0, and it showcases a natural combination of W-trick methods with Harper's restriction estimates for smooth numbers. The manuscript is clearly organized and openly relies on external results of Harper, Green, and Green–Tao; I see no circularity or parameter fitting. However, several load-bearing steps are not justified as written, and the current proof does not establish the theorem. The main ideas are plausible and a repair may be possible, but the gaps are substantial and concern the central argument.","major_comments":[{"comment":"Proposition 3.6 explicitly requires N to be a large prime, but the proof applies it with modulus N_b = floor(N/(b_1 W)) + 1, which is generally composite. No composite-modulus analogue of the transference principle, and no reduction to a prime modulus (e.g., by embedding A_b into Z/PZ for a prime P > 3N_b), is supplied. Since the lower bound for the f_b-triple correlation is the central input, this is a load-bearing gap.","section":"§4, application of Proposition 3.6"},{"comment":"Even if a valid transference statement existed on Z/N_b Z, the resulting count is a count of arithmetic progressions modulo N_b. A_b is a subset of [N_b], and a modular progression such as (N_b-1, 0, 1) or, for example, (4, 1, 3) in Z/5Z can wrap around and does not correspond to an integer arithmetic progression in [N_b]. The concluding sentence, which says that a progression with difference d in A_b corresponds to a progression with difference b_1 W d in A, depends on having an integer progression. A standard repair is to work on a prime P > 3N_b and restrict the support to [1, P/3]; the paper does not carry this out, and Propositions 4.2–4.3 are stated only for modulus N_b.","section":"§4, final paragraph: modular vs integer progressions"},{"comment":"Theorem 3.7, Harper's restriction theorem, is stated for exponential sums over y-smooth numbers n ≤ x, i.e., n ∈ S(y). In the proof of Proposition 4.3 the L^p bound is applied to the sum over n ∈ A_b, but A_b has not been shown to consist of y-smooth numbers. Smoothness of b_1(W n − b_2) does not imply smoothness of n. Thus the bound N_b^{α p + o(1)} is not justified, and the estimate M = W^{p(1−α)} used in the transference is unsupported.","section":"§5, proof of Proposition 4.3"},{"comment":"The statement of Proposition 4.2 concerns the exponential sum with e(a n / N_b) over n ∈ Z/N_b Z. The proof, however, estimates sums of the form e(a' n / N) and uses the major-arc/minor-arc decomposition of §3.2, which is defined with error R/N where R = log^20 N. Since N_b can be as small as N^{1/2+o(1)}, the rational approximation for θ = a/N_b with denominator q ≤ log^20 N is not generally available; the proof does not establish the claimed o(1) for the normalized Fourier coefficients. This is another load-bearing issue because Proposition 4.2 is one of the three hypotheses needed for the transference principle.","section":"§5, proof of Proposition 4.2"}],"minor_comments":[{"comment":"In the displayed conclusion, 'f(n+d)f(n+d)' should presumably read 'f(n+d)f(n+2d)'.","section":"§3.3, Proposition 3.6"},{"comment":"The constants are not fully tracked: Proposition 4.1 gives the average of f_b as δ^{2−α}, while Proposition 3.6 is stated with a parameter δ. The proof says 'with η = o(1), M = W^{p(1−α)}' but does not explicitly define the new density parameter or verify all hypotheses of Proposition 3.6 in the notation of that proposition.","section":"§4, proof of Theorem 1.1"},{"comment":"The displayed chain '≫ δ^{2−α} N^{α+o(1)}_b W^{1−α} W^{1−α} N^{α+o(1)}_b = δ^{2−α} N_b' is difficult to follow and appears to contain redundant factors; the intermediate estimates should be rewritten with the relation N_b ≈ N/(b_1 W) made explicit.","section":"§5, proof of Proposition 4.1"},{"comment":"The phrase 'it is more convenient to us [sic] to normalize' contains a typo. More importantly, the proof uses a'/N in the exponential while the proposition requires a/N_b; the notation should be made consistent.","section":"§5, proof of Proposition 4.2"},{"comment":"The phrase 'under a weaker hypothesis' is ambiguous: Harper allows y as small as log^K x with K depending on δ, while here K is fixed and the theorem is stated for all δ; the relation between the two hypotheses could be stated more precisely.","section":"§1, abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a clear and plausible strategy, and the citations to Harper, Green, and Green–Tao are appropriate. But the proof as written has four unresolved technical gaps: the composite modulus in the transference, the modular-to-integer AP conversion, the application of Harper's restriction theorem to a non-smooth set A_b, and the mismatch in Proposition 4.2's exponential sums. These are not mere presentation issues; they affect the central claim. I would not recommend acceptance in the current form, but the approach may be repairable, especially if the transference is done on a large prime modulus with support in [1, P/3] and the restriction-theoretic input is reworked for the actual set A_b."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper likely proves the right theorem—fixed K in Roth's theorem for super smooth numbers—and the strategy is appropriate: W-trick, Green-Tao transference, and Harper's exponential sum estimates. The fixed-K statement is genuinely new; Harper only had K depending on delta, and the introduction notes he suggested the W-trick would give fixed K but did not carry it out. That makes this a worthwhile extension.\n\nWhat the paper does well: it adapts the W-trick to the smooth numbers setting, uses the right major/minor arc decomposition, and keeps the exposition clear. The dependence on Harper's machinery is explicit and not circular.\n\nBut there are three soft spots, and they are in the load-bearing core. First, Proposition 3.6 is the Green-Tao transference principle for a large prime modulus N, yet the proof applies it to N_b = floor(N/(b_1 W))+1, which is generally composite. The Fourier analysis on Z/N_b Z is not automatically equivalent to the circle method on [1,N_b], and no composite-modulus analogue or reduction to a prime is supplied. Second, even if the transference worked, the count on Z/N_b Z is a count of modular 3-APs. These can wrap around the modulus, and the final step converting them to integer APs in A_b is invalid without a modulus larger than 3N_b or another no-wrap argument. The example of residues 4,1,3 in Z/5Z shows the issue: they form a modular AP but not an integer AP. Third, Proposition 4.3 invokes Harper's restriction theorem on the exponential sum over A_b, but A_b is not a set of y-smooth numbers; the theorem's hypothesis on the support is not met. Each of these is fixable—choose a prime P > 3N_b and work on Z/PZ, or prove the needed restriction estimate for the actual set A_b—but none of the fixes is in the paper.\n\nThe minor issues in Proposition 4.2 and Lemma 5.1 are cosmetic by comparison. The core idea is sound and the gaps are technical rather than conceptual. I would not desk-reject this; it deserves a serious referee who can help the author close these gaps. If repaired, it is a solid contribution to the smooth-numbers literature.","headline":"Plausible result, right strategy, but the proof is incomplete: the transference is applied on composite moduli and the modular AP count is not converted to integer APs; these are load-bearing and need repair.","tokens_in":657,"tokens_out":778,"would_cite":false,"duration_ms":45434,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B25","11L07","11N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every relatively dense subset of super smooth numbers contains a nontrivial 3-term arithmetic progression, for any fixed large smoothness exponent K.","keywords":["arithmetic progressions","Roth's theorem","smooth numbers","super smooth numbers","W-trick","transference principle","exponential sums","restriction estimates"],"falsifier":"Check Proposition 4.2 at a composite modulus: for N_b even, compute (1/N_b) Σ_{n} ν_b(n) e(n N_b / 2) for the W-trick weight with b1 = 1. The transference principle's pseudorandomness condition requires this quantity to be o(1); if it is bounded away from zero for infinitely many N, the Fourier-decay step fails and the proof cannot stand as written.","tokens_in":8897,"feed_emoji":"🔢","tokens_out":6332,"duration_ms":52408,"temperature":0.7,"pith_summary":"The paper aims to establish Roth's theorem for super smooth numbers: if y = log^K N with K a fixed large integer, and A is a subset of the y-smooth numbers up to N with size at least δ times the total count of such numbers, then A contains a nontrivial arithmetic progression of length 3. This extends a previous result that required K to grow as δ goes to zero, to the case where K is merely a large constant. A sympathetic reader would care because the smooth numbers are extremely sparse—about N^{1-1/K}—yet the theorem asserts they still have the same 3-progression rigidity as dense sets. The proof works by a W-trick that isolates a dense-enough molded copy of the set, then uses a transference principle to recover the arithmetic progression.","feed_headline":"Fixed-K smooth numbers force 3-term arithmetic progressions","feed_subtitle":"A W-trick transfer shows every relatively dense subset of log^K N-smooth numbers contains a nontrivial 3-AP.","key_machinery":"The W-trick weight ν_b(n) = C_W (W n - b_2)^{1-α} 1_{b_1(W n - b_2)∈S(N,y)}, supported on n ≤ N_b, with C_W = ∏_{p|W} (1 - p^{-1})/(1 - p^{-α}) and α = 1 - 1/K + o(1). It does three jobs: it has l^1 mass asymptotic to N_b, so it models the smooth set in the W-tricked world; its nonzero Fourier coefficients are o(1), making it pseudorandom enough for transference; and the restricted function f_b = 1_{A_b}ν_b satisfies the p-th moment bound Σ_a |(1/N_b)Σ_n f_b(n)e(an/N_b)|^p ≪ W^{p(1-α)}. These three properties are exactly what the transference principle needs to output a triple-correlation lower bound.","core_discovery":"Theorem 1.1 states that for any fixed large K, any δ > 0, and all large N in terms of δ and y = log^K N, every subset A of the y-smooth numbers up to N with |A| ≥ δΨ(N, y) contains a nontrivial 3-term arithmetic progression. The proof passes to a W-tricked set A_b, constructs a normalized weight ν_b supported on numbers whose b1(Wn - b2) form is smooth, and shows that the triple correlation of the modified characteristic function f_b = 1_{A_b}ν_b is bounded below by a positive constant. The weight has total mass ≈ N_b, Fourier coefficients o(1) away from zero, and a controlled l^p moment. Applying the transference principle then gives a lower bound for the triple correlation of 1_{A_b}; any","pith_inferences":["If the composite-modulus gap in the transference step can be closed by a proper circle-method reduction, the same framework would yield a fully self-contained proof and possibly explicit bounds on how large N must be in terms of δ and K.","The estimates seem to rely only on α > 1/2 (i.e., K > 2), so the 'large K' hypothesis might be replaceable by K ≥ 3 with a more careful constant chase.","The W-trick weight construction and the pattern of the proof are adapted from the squarefull-numbers case, suggesting the method may transfer to other sparse multiplicative sets whose exponential sums satisfy similar minor-arc and restriction bounds.","It would be natural to test numerically whether the composite-modulus issue actually breaks the claimed Fourier decay for N_b even: a single counterexample to Proposition 4.2 at a composite modulus would indicate where a repair is needed."],"forward_implications":["Theorem 1.1 settles the fixed-K case: for y = log^K N with K a fixed large integer, every δ-dense subset of y-smooth numbers contains a nontrivial 3-term arithmetic progression for all sufficiently large N.","The 3-progression found in the W-tricked set A_b lifts to a 3-progression in the original set A with common difference b1 W d, so the W-trick genuinely transfers configurations back, not just density.","The Fourier estimates for ν_b (Proposition 4.2) and the restriction estimate for smooth numbers (Theorem 3.7) provide a reusable package for other additive problems over smooth numbers.","The proof yields a quantitative lower bound on the number of 3-progressions, of shape ≫ N_b^{3α-1}/(C_W^3 W^{3-3α}) - N_b^{α+o(1)}, showing the existence statement is in principle effective.","Because the argument works for any fixed large K, it covers the full 'super smooth' range y = log^K N, which was explicitly left open by earlier approaches that required K to grow with 1/δ."],"fun_headline_variants":["Roth's theorem holds for log-K smooth numbers","Super smooth numbers contain 3-term APs","Dense smooth subsets force 3-APs via W-trick","Power-log smooth numbers pass Roth's test"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof applies a transference principle that explicitly requires the ambient modulus to be a large prime, but N_b = floor(N/(b1 W)) + 1 is generally composite; the paper does not reduce to a prime modulus nor supply a composite-modulus analogue, and the lower bound for the triple correlation depends on this step.","fun_headline_variants_meta":{"raw":{"variants":["Roth's theorem holds for log-K smooth numbers","Super smooth numbers contain 3-term APs","Dense smooth subsets force 3-APs via W-trick","Power-log smooth numbers pass Roth's test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1040,"prompt_tokens":621,"completion_tokens":419,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":355}},"tokens_in":365,"tokens_out":419,"duration_ms":4192,"temperature":1.0,"reasoning_tokens":355,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:55:30.571522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check Proposition 4.2 at a composite modulus: for N_b even, compute (1/N_b) Σ_{n} ν_b(n) e(n N_b / 2) for the W-trick weight with b1 = 1. The transference principle's pseudorandomness condition requires this quantity to be o(1); if it is bounded away from zero for infinitely many N, the Fourier-decay step fails and the proof cannot stand as written.","supporting_citations":[],"review_version":1}