{"id":"82224f32-8141-4c0a-a9d8-6d20d9ae9f09","arxiv_id":"2411.16069","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For dense enough subsets A,B of (N,2N], there exist a in A and b in B with ab = P_k^2 + O(P_k^{1-delta}) for an almost-prime P_k.","lead":"This paper shows that two large number sets A and B, each of a prescribed size, always contain a from A and b from B whose product ab is very close to the square of an almost-prime, a number with only a bounded number of prime factors. It refines the classical near-square theorem of Iwaniec and Sarkozy by using exponential sum estimates inside a linear sieve.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 is false as stated: (1.4) defines A as a set of at most N distinct integers, while (3.14) counts pairs, so for dense A,B the asserted lower bound exceeds |A|.","rationale":"The deepest issue in the paper is the mismatch between the set A defined in (1.4) and the multiset implicitly used in the sieve. This is load-bearing because every subsequent sifting statement, in particular (3.14) and the lower bound in Theorem 1.2, depends on treating the rounded square roots with multiplicity one per pair (a,b). If A is read literally as the set of distinct rounded roots, Theorem 1.2 is not merely unproved but false: the asserted lower bound can exceed the total number of available integers l. The reader's verdict correctly identified this as the weakest assumption. I do not see a comparable flaw in the exponential sum estimates of Lemma 3.1: the dyadic decomposition and the choice H=dΔ^{-1}N^{2ε} are consistent with the stated density condition, and the bound (3.23) appears sufficient. The numerical positivity asserted in Theorem 1.5 should ideally be backed by reproducible code or a written verification, but that is a minor issue compared with the set-versus-multiset defect. Since the central existence claim can be recovered by an explicit multiset reinterpretation, the appropriate verdict remains conditional rather than an outright rejection: the authors should revise the definitions and theorem statements so that the counting object is unambiguously the multiset of pairs, or prove the distinct-set version with a genuinely different argument.","tokens_in":1174,"tokens_out":889,"duration_ms":222837,"concrete_test":"Check the cardinality of the derived set for the extreme dense case A=B={N+1,...,2N}, Δ=N^{-δ} with δ=1/20. The set A in (1.4) has at most N+1 elements (all in [N,2N]), while Theorem 1.2 predicts H(A;k) ≥ C N^{2-δ}/logN, which exceeds N for large N. Since H(A;k) ≤ |A|, this is a contradiction. Replacing the set in (1.4) by a multiset and redefining H as a count of pairs removes the contradiction; the numerical sieve estimates in Sections 3 and 4 then apply as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central counting step conflates a set with a multiset. In (1.4), A is defined as a set of closest integers l(√ab) obtained from pairs (a,b) satisfying ||√ab||<Δ. Since a,b∈[N+1,2N], every l lies in [N,2N], so the derived set has at most N+1 elements. Nevertheless, (3.14) computes |A_d| as Σ_{a∈A}Σ_{b∈B}(⌊(√ab+Δ)/d⌋-⌊(√ab-Δ)/d⌋), i.e. it treats each pair as contributing a distinct element. This equality is false when two pairs share the same rounded root. Consequently the lower bound in Theorem 1.2, H(A;k) ≥ C(η,β,δ) Δ|A||B|/logN, can exceed the maximum possible H(A;k). For example, with A=B={N+1,...,2N}, η=β=1 and δ=1/20, the RHS is ≍ N^{2-δ}/logN, while H(A;k) ≤ |A| ≤ N+1. Thus the literal theorem is false; the proof actually establishes a lower bound for pairs, not for distinct rounded roots. The fix is to define the sieve sequence as a multiset (or to replace H(A;k) by a count of pairs) and to state Theorem 1.2 accordingly. As written, the existence corollary rests on an unstated multiset interpretation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a multiplicative hybrid problem: for dense subsets A and B of {N+1,...,2N}, it seeks a ∈ A and b ∈ B such that ab is close to the square of an almost-prime. The main result, Theorem 1.2, claims a lower bound of order Δ|A||B|/log N for the count of closest integers l(√ab) that have few prime factors, where the l's are collected in the set A defined in (1.4). Theorem 1.5 refines this to k = 4 and 5 using a weighted sieve, and an application (Corollary 1.3) asserts the existence of such a pair. The proofs combine the linear sieve, exponential-sum estimates of Fouvry-Iwaniec, and the Halberstam-Richert sieve framework, with numerical integrations used to certify positivity of constants.","tokens_in":12671,"tokens_out":15635,"duration_ms":131914,"significance":"The result is a natural near-prime refinement of the Iwaniec-Sárközy multiplicative hybrid theorem, and the existence corollary, if secured, would be a meaningful step toward the conjectured prime-square version. The paper is not circular: it uses established external estimates and does not fit constants to the target result, and the sieve computations are explicit. However, the set-versus-multiset confusion in the definition of A makes Theorem 1.2 false as stated, and the proof of the crucial uniformity estimate in Lemma 3.1 has a gap for small dyadic blocks. These issues are load-bearing: without a multiset reinterpretation or a distinct-element argument, the claimed lower bound for the set A does not follow. The numerical positivity assertions in Section 5 also need rigorous certification before Theorem 1.5 can be accepted.","major_comments":[{"comment":"The displayed chain |A_d| = ∑_{l∈A, d|l} 1 = ∑_{a∈A,b∈B, |√ab-l|<Δ, d|l} 1 = ∑_{a∈A}∑_{b∈B}( ⌊(√ab+Δ)/d⌋ - ⌊(√ab-Δ)/d⌋ ) is valid only if the map (a,b) ↦ l(√ab) is injective on the set of pairs satisfying ||√ab||<Δ. Since A is defined in (1.4) as the set of distinct closest integers, the third equality counts each rounded root once for every pair that produces it, and the right-hand sum can strictly exceed |A_d|. Thus the sieve sequence underlying Lemma 3.1 and Theorem 1.2 is a multiset with multiplicity indexed by pairs (a,b), not the set A of distinct l. This conflation is the root cause of the failure of Theorem 1.2 as stated.","section":"Section 3, Eq. (3.14)"},{"comment":"As stated, Theorem 1.2 is false. For η=β=1, δ=1/20, take A=B={N+1,...,2N}. Then |A|=|B|=N, Δ=N^{-1/20}, and the asserted lower bound is of order N^{2-δ}/log N ≫ N^{39/20}, while H(A;k) ≤ |A| ≤ N+1 because A⊂{N,...,2N}. Hence the lower bound exceeds the maximum possible value of H(A;k) for all sufficiently large N. The proof actually establishes a lower bound for the number of pairs (a,b) such that l(√ab) has at most k prime factors counted with the multiplicity of pairs. Theorem 1.2 and the surrounding definitions must be restated either for the multiset of rounded roots with one copy per pair, or one must supply an additional argument passing from pair counts to distinct integers l. Corollary 1.3 may survive the pair-counting restatement, but it cannot be derived from Theorem 1.2 as written.","section":"Theorem 1.2"},{"comment":"The uniformity claim r(A,d) ≪ X N^{-ε}/d for each 1 ≤ d ≤ X^α is not established by the displayed estimates. In the dyadic decomposition after (3.18), the parameter H1 ranges over values as small as O(1) (since H1 = 2^{-j-1}H with j up to log H). For such a block, (3.23) gives M ≪ N(|A||B|)^{1/4}(1 + d^{1/2}H_1^{-1/2}) log^{1/2}N, and after multiplication by log H/H1 the second term in (3.24) is of size comparable to N^{1+(η+β)/4} d^{1/2} log^{3/2}N. With d = X^α, the exponent of this term equals the exponent of d^{-1}X exactly at the boundary η+β = 4(1+δ)/3 + ε, so only a logarithmic saving is obtained, not the required N^{-ε}. The small dyadic blocks therefore break the claimed N^{-ε} saving, and the bound (4.2) used to verify the sieve condition (R(1,α)) does not follow. This gap is load-bearing for the lower bounds in Theorems 1.2 and 1.5.","section":"Lemma 3.1"},{"comment":"The positivity of C(δ,5) for 0<δ<1/10 and the numerical lower bound C(δ,4)>0.0023205 for 0<δ<121/10000 are asserted from 'Use Mathematica for numerical computation.' No rigorous error bounds for the numerical integrations are provided. Since Theorem 1.5 rests precisely on these inequalities, the authors should either supply certified interval-arithmetic bounds or an analytic proof of positivity on the stated ranges; a floating-point assertion is not sufficient for a formal proof.","section":"Section 5, Eq. (5.10)-(5.11)"}],"minor_comments":[{"comment":"The abstract uses 0<c<1/2 but all theorems state 0<δ<1/2; the notation should be unified.","section":"Abstract / Introduction"},{"comment":"In the line 'recalling the definition (3.19)', the reference should be to (1.4), not (3.19).","section":"Section 3"},{"comment":"The sifting function S(A,z) is defined for a set A, but after the multiset interpretation introduced by (3.14), it should be made explicit whether multiplicities are counted; otherwise the reader cannot tell whether (4.1) and (4.3) are counting distinct elements or pairs.","section":"Sections 2-4"},{"comment":"The statement would benefit from an explicit condition making the main term dominant, e.g., Δ not too small relative to |A||B|; currently the error term N(|A||B|)^{1/4} log^{3/2}N can exceed the main term for very small Δ, and the theorem is only meaningful in the range δ < 3(η+β)/4 - 1.","section":"Theorem 1.1"},{"comment":"There are minor typographical issues, such as 'T echnology' in the second affiliation and inconsistent use of the closest-integer notation; a careful proofread is recommended.","section":"Affiliation / Formatting"}],"recommendation":"major_revision","confidential_remarks":"The central existence result may well be salvageable by restating Theorem 1.2 for the multiset of rounded roots indexed by pairs (a,b), which would preserve Corollary 1.3. However, the current set-based statement is false, and the uniformity gap in Lemma 3.1 is a substantive technical problem that requires new estimates or a different treatment of small dyadic blocks. The numerical positivity in Section 5 should also be made rigorous. I recommend major revision rather than rejection, because the overall strategy is plausible and the requested changes, though nontrivial, are within the scope of a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read. The genuinely new thing is trying to make ab land near a square of an almost-prime rather than near a square of an arbitrary integer, and to get there by sieving the rounded square roots with the linear sieve after estimating the discrepancy with Fouvry–Iwaniec exponential sums. The plan is natural and, modulo one definitional slip, the proof looks sound to me. There is no fitting and no circularity; the cited benchmarks are external and the constants come from explicit integrals.\n\nThe slip is load-bearing. Equation (1.4) defines A as a set of distinct closest integers l(√ab). Since l ∈ [N,2N], |A| ≤ N+1. But (3.14) evaluates |A_d| by summing over pairs (a,b), i.e., it treats A as a multiset with one copy per pair. The equality is false when two pairs share the same rounded root. Consequently Theorem 1.2 is false as stated: with |A|=|B|=N and small δ, the claimed lower bound is of order N^{2−δ}/log N, which exceeds the total number of available l. The proof actually establishes a lower bound for the number of pairs (a,b), not for the number of distinct l. This is fixable: restate the object as a multiset or replace H(A;k) by the pair count, and the sieve argument goes through. As written, the corollary rests on an unstated multiset interpretation.\n\nThe other issues are minor. In Theorem 1.5 the positivity of C(δ,k) is verified by \"Use Mathematica\" without code; the formula is explicit, so this is a reproducibility complaint, not a mathematical gap. Also, Lemma 2.3 as printed gives a weaker bound than the one used in (3.22), but the needed bound follows directly from integrality of b, so I read that as a typo.\n\nWho is this for? People working on multiplicative hybrid problems and sieve methods. It is a solid subfield contribution if corrected, not a breakthrough. I would not cite the current arXiv version in my own work, but I would cite a corrected version. I would send it to a serious referee, with instructions that the statement must be fixed before acceptance.","headline":"The intended result is new and plausible, but Theorem 1.2 is false as stated because the proof counts pairs while the notation defines a set of distinct rounded square roots.","tokens_in":13140,"tokens_out":14823,"would_cite":false,"duration_ms":129202,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N36","11L07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two sufficiently dense subsets of an interval always contain a product within O(P_k^{1-δ}) of the square of an almost-prime with at most k prime factors.","keywords":["linear sieve","almost-prime","multiplicative hybrid problem","exponential sums","sifting function","near-square","weighted sieve","rounded square root"],"falsifier":"Take N moderately large, choose A and B as full intervals, pick d near X^α and H=$dΔ^{{-1}}$$N^{{2ε}}$, and compute the exponential sum $S=\\sum_{a\\in A}\\sum_{b\\in B} e(h\\sqrt{ab}/d)$ for h∼H. If |S| exceeds the bound in (3.23) by a positive power of N, then the uniform estimate behind Lemma 3.1 is false and the stated lower bound does not follow. Alternatively, compute |r(A,d)| directly for such d and check whether it remains ≤ $XN^{{-ε}}$/d.","tokens_in":12089,"feed_emoji":"🔢","tokens_out":6766,"duration_ms":58477,"temperature":0.7,"pith_summary":"This paper proves that two large subsets A and B of {N+1,...,2N} can always be multiplied to land close to the square of an almost-prime: there exist a∈A and b∈B with ab = $P_k^{2}$ + O($P_k^{{1-δ}}$), where P_k has at most k prime factors and k is determined by the densities of A and B. The denser the sets, the smaller the allowable error and the fewer prime factors are needed; when both sets have size comparable to N, the result holds for every δ<1/14 with k=6. Earlier work in this direction only guaranteed that some product ab is near a square. This paper refines that conclusion by showing the nearby square can be the square of a number with a bounded number of prime factors, and it also gives a quantitative lower bound on how many pairs (a,b) actually work.","feed_headline":"Two dense sets always multiply close to an almost-prime square","feed_subtitle":"New sieve argument shows such products exist with at most six prime factors, improving the old near-square guarantee.","key_machinery":"The load-bearing mechanism is the classical linear sieve applied to the collection of nearest integers l(√ab) obtained from pairs (a,b) with ‖√ab−l‖<Δ. The count of such l divisible by d is written as $|A_d| = (2\\Delta |A||B|)/d + r(A,d)$, with the error $r(A,d)$ controlled by Lemma 3.1. That lemma estimates the error by approximating the fractional-part function ψ via short exponential sums and by bounding double exponential sums of the form $\\sum_{a,b} e(h\\sqrt{ab}/d)$, using dyadic decomposition and a second-moment inequality for exponential sums with monomials. The sieve then produces a lower bound for the sifting function $S(A,(3N)^{1/(k+1)})$, and any integer surviving that sifting has no prime factor below $(3N)^{1/(k+1)}$, hence has at most k prime factors. The k=4,5 cases replace the plain sieve with a weighted sieve to obtain a positive constant over a narrow range of δ.","core_discovery":"The central assertion is Theorem 1.2 and its Corollary 1.3: for A,B⊂{N+1,...,2N} with |A|≍N^η, |B|≍N^β, and η+β ≥ 4(1+δ)/3+ε, the number of pairs (a,b) for which the nearest integer l to √(ab) has at most k prime factors is at least C(η,β,δ) Δ|A||B|/log N, where Δ=$N^{{-δ}}$ and k=⌊2/((η+β)/2-2/3-2δ/3)⌋. In particular, at least one such pair exists, yielding ab=$P_k^{2}$+O($P_k^{{1-δ}}$). The paper also proves stronger small-k versions: if |A|,|B|≫$N^{{1-ε}}$, the same conclusion holds for k=5 with 0<δ<1/10, and for k=4 with 0<δ<121/10000. This is an approximate realization of the conjecture that a prime square (k=1) should suffice.","pith_inferences":["Read literally, the set A defined in (1.4) collects distinct rounded square roots, while the counting in Theorem 1.2 needs one copy of l(√ab) for each pair (a,b); if A is not treated as a multiset, the claimed lower bound can exceed the number of distinct elements. The proof is most charitably read as a statement about pairs (a,b).","If the exponential sum estimates behind Lemma 3.1 could be strengthened uniformly in d, the admissible ranges of δ would widen, potentially moving toward the conjectured prime-square case k=1.","The method depends only on the sizes of A and B, not their arithmetic structure, so a similar sieve-plus-exponential-sum strategy might apply to other sparse subsequences, such as polynomial values, if an analogue of Lemma 3.1 can be established."],"forward_implications":["When both A and B have size comparable to N, every admissible pair of sets yields some product ab within N^{-δ} of a number with at most 6 prime factors, for every δ<1/14.","The quantitative bound $H(A;k) \\gg \\Delta |A||B|/\\log N$ shows that many pairs survive the sieve, not just one pair.","For very dense sets, |A|,|B|≫N^{1-ε}, the number of prime factors can be reduced to 4 or 5 at the cost of a smaller allowable error δ.","The threshold condition η+β ≥ 4(1+δ)/3+ε marks a density boundary: if both sets are too sparse, the present method no longer forces any near almost-prime square.","The result directly extends the earlier near-square theorem: it shows the near-square phenomenon is stable under replacing 'square' by 'almost-prime square' with a bounded number of prime factors."],"supporting_citations":[{"why":"Establishes the original near-square multiplicative hybrid problem and supplies the lemma controlling the number of close pairs of square roots in B.","marker":"[5]"},{"why":"Provides the exponential sum lemmas used to estimate the error term r(A,d) in Lemma 3.1.","marker":"[2]"},{"why":"Supplies the linear sieve lower and upper bounds used to obtain the main sifting estimate for S(A,z).","marker":"[4]"},{"why":"Gives the exponential sum approximation of the fractional-part function ψ used to separate the main term from the error.","marker":"[7]"},{"why":"Provides the weighted sieve used to derive the k=4,5 cases in Theorem 1.5.","marker":"[6]"}],"fun_headline_variants":["Dense sets guarantee products near an almost-prime square","Sieve method shows dense sets multiply to almost-prime square","Almost-prime square guaranteed for product of two dense sets","Product of dense sets lands near an almost-prime square","Two dense sets force a product near an almost-prime square"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on Lemma 3.1: for every d up to X^α the error r(A,d) is bounded by X $N^{{-ε}}$/d, with the saving coming from two exponential-sum lemmas and from choosing the truncation H=$dΔ^{{-1}}$$N^{{2ε}}$; if that uniform error bound fails for some d in the range, the sieve lower bound collapses.","fun_headline_variants_meta":{"raw":{"variants":["Dense sets guarantee products near an almost-prime square","Sieve method shows dense sets multiply to almost-prime square","Almost-prime square guaranteed for product of two dense sets","Product of dense sets lands near an almost-prime square","Two dense sets force a product near an almost-prime square"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001063,"raw_usage":{"total_tokens":4408,"prompt_tokens":849,"completion_tokens":3559,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":3476}},"tokens_in":465,"tokens_out":3559,"duration_ms":22960,"temperature":1.0,"reasoning_tokens":3476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:36:15.553974+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take N moderately large, choose A and B as full intervals, pick d near X^α and H=$dΔ^{{-1}}$$N^{{2ε}}$, and compute the exponential sum $S=\\sum_{a\\in A}\\sum_{b\\in B} e(h\\sqrt{ab}/d)$ for h∼H. If |S| exceeds the bound in (3.23) by a positive power of N, then the uniform estimate behind Lemma 3.1 is false and the stated lower bound does not follow. Alternatively, compute |r(A,d)| directly for such d and check whether it remains ≤ $XN^{{-ε}}$/d.","supporting_citations":[{"cited_title":"Iwaniec and A","cited_arxiv_id":null,"evidence_quote":"Establishes the original near-square multiplicative hybrid problem and supplies the lemma controlling the number of close pairs of square roots in B."},{"cited_title":"Fouvry and H","cited_arxiv_id":null,"evidence_quote":"Provides the exponential sum lemmas used to estimate the error term r(A,d) in Lemma 3.1."},{"cited_title":"Halberstam and H","cited_arxiv_id":null,"evidence_quote":"Supplies the linear sieve lower and upper bounds used to obtain the main sifting estimate for S(A,z)."},{"cited_title":"Rivat and A","cited_arxiv_id":null,"evidence_quote":"Gives the exponential sum approximation of the fractional-part function ψ used to separate the main term from the error."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weighted sieve used to derive the k=4,5 cases in Theorem 1.5."}],"review_version":1}