On a multiplicative hybrid problem over almost-primes

Wenguang Zhai; Yuetong Zhao

REVIEW 4 major objections 5 minor 8 references

On a multiplicative hybrid problem over almost-primes

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.16069 v1 pith:V3EVYZ43 submitted 2024-11-25 math.NT

Yuetong Zhao , Wenguang Zhai This is my paper · ORCID

classification math.NT MSC 11N3611L07

keywords linearsievealmost-primemultiplicativehybridproblemexponentialsumssiftingfunctionnear-squareweightedroundedsquareroot

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[Section 3, Eq. (3.14)] 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.
[Theorem 1.2] 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.
[Lemma 3.1] 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 5, Eq. (5.10)-(5.11)] 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.

minor comments (5)

[Abstract / Introduction] The abstract uses 0<c<1/2 but all theorems state 0<δ<1/2; the notation should be unified.
[Section 3] In the line 'recalling the definition (3.19)', the reference should be to (1.4), not (3.19).
[Sections 2-4] 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.
[Theorem 1.1] 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.
[Affiliation / Formatting] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main count is an application of external exponential-sum and sieve lemmas with explicitly computed constants, not an equivalence to its inputs.

full rationale

Circularity score 0. The derivation chain does not reduce to its inputs. Theorem 1.1 estimates H(A,B;Delta) by expanding the floor difference via the Vaaler-type Fourier expansion (Lemma 2.7 from Rivat and Sarkozy) and bounding the resulting exponential sums with the Fouvry-Iwaniec bilinear-form lemmas; the main term 2Delta|A||B| is the expected value, not a fitted quantity. Lemma 3.1 controls the sieve remainders r(A,d) using the same external exponential-sum estimates and the explicit parameter choice H = d*Delta^{-1}N^{2epsilon}, and this remainder bound is then fed into the Halberstam-Richert linear sieve (Lemma 2.8) with multiplicative density omega(d)=1. The lower-bound constant C(eta,beta,delta) in Theorem 1.2 comes from the explicit sieve functions F and f of Lemma 2.4; Theorem 1.5's constants C(delta,k) are evaluated from specified integrals and numerical computation, with no parameter tuned to force the target. The integer k is chosen exactly so that f(alpha(k+1)(eta+beta-delta))>0, a positivity condition, not a fitted prediction. No load-bearing self-citation occurs: reference [8] is the earlier result being generalized, and reference [6] is a textbook sieve source. The set-versus-multiset ambiguity between definition (1.4) and equation (3.14), noted in the reader's take, is a correctness concern rather than circularity: it does not make any claimed output equal to an input by construction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof's central claim rests on standard sieve and exponential sum results plus an unflagged multiset reinterpretation of A. There are no fitted free parameters or invented entities.

assumptions (5)

standard math Fouvry-Iwaniec exponential sum bounds (Lemmas 2.1, 2.2) are valid as stated.
Used in the proof of Lemma 3.1 to estimate D1 and D2; cited from [2].
standard math Vaaler approximation of psi(t) (Lemma 2.7) is valid.
Provides the Fourier expansion of the sawtooth function used throughout; cited from [7].
standard math Linear sieve lower and upper bounds (Lemmas 2.8, 2.9) apply to the sequence A with omega(d)=1 and dimension 1.
The sieve theorems from Halberstam-Richert are used to lower bound S(A,z).
ad hoc to paper The sequence of rounded roots can be treated as a multiset for the sieve, even though A is defined as a set.
The identity (3.14) counts each pair (a,b) separately; if the same l arises from several pairs, the count is not the cardinality of the set A. This is unflagged in the paper.
ad hoc to paper The numerical integrations in Section 5 yield C(delta,5)>0 for delta<1/10 and C(delta,4)>0.0023205 for delta<121/10000.
The constants are asserted after 'Use Mathematica for numerical computation' without providing code or a proof of positivity.

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Cite this review

Pith. "Pith review of On a multiplicative hybrid problem over almost-primes." pith.science (2026). https://pith.science/paper/V3EVYZ43

@misc{pith2026241116069,
  author       = {Pith},
  title        = {Pith review of: On a multiplicative hybrid problem over almost-primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3EVYZ43}},
  note         = {Machine review of arXiv:2411.16069}
}

read the original abstract

Let N be a large enough natural number, A and B be subsets of {N+1, ... , 2N}. In this paper, we prove that there exists integers a, b with a belongs to A, b belongs to B such that ab=P_k^2 + O(P_k^{1-c}), where 0<c<1/2 and P_k denotes an almost-prime with at most k prime factors, counted with multiplicity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Bordell` es, A note on a multiplicative hybrid problem, Acta Arith

O. Bordell` es, A note on a multiplicative hybrid problem, Acta Arith. 134 (4) (2008) 387– 393

work page 2008
[2]

Fouvry and H

E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989) 311–333

work page 1989
[3]

S. W. Graham and G. Kolesnik, Van der Corput’s Method of Expone ntial Sums, Cambridge University Press, New York (1991)

work page 1991
[4]

Halberstam and H

H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, London (1974)

work page 1974
[5]

Iwaniec and A

H. Iwaniec and A. S´ ark¨ ozy, On a multiplicative hybrid problem, J. Number Theory 26 (1987) 89–95

work page 1987
[6]

C. D. Pan and C. B. Pan, Goldbach Conjecture, Beijing, Science P ress, 1981 (in Chinese)

work page 1981
[7]

Rivat and A

J. Rivat and A. S´ ark¨ ozy, A sequences analog of the Piatetski-Shapiro problem, Acta Math. Hungar. 74 (3) (1997) 245–260

work page 1997
[8]

W. G. Zhai, On a multiplicative hybrid problem, Acta Arith. 71 (1995) 47–53 15

work page 1995

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bordell` es, A note on a multiplicative hybrid problem, Acta Arith

O. Bordell` es, A note on a multiplicative hybrid problem, Acta Arith. 134 (4) (2008) 387– 393

work page 2008

[2] [2]

Fouvry and H

E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989) 311–333

work page 1989

[3] [3]

S. W. Graham and G. Kolesnik, Van der Corput’s Method of Expone ntial Sums, Cambridge University Press, New York (1991)

work page 1991

[4] [4]

Halberstam and H

H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, London (1974)

work page 1974

[5] [5]

Iwaniec and A

H. Iwaniec and A. S´ ark¨ ozy, On a multiplicative hybrid problem, J. Number Theory 26 (1987) 89–95

work page 1987

[6] [6]

C. D. Pan and C. B. Pan, Goldbach Conjecture, Beijing, Science P ress, 1981 (in Chinese)

work page 1981

[7] [7]

Rivat and A

J. Rivat and A. S´ ark¨ ozy, A sequences analog of the Piatetski-Shapiro problem, Acta Math. Hungar. 74 (3) (1997) 245–260

work page 1997

[8] [8]

W. G. Zhai, On a multiplicative hybrid problem, Acta Arith. 71 (1995) 47–53 15

work page 1995