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REVIEW 4 major objections 5 minor 21 references

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.

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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.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection 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. the 4 major comments →

arxiv 2510.18024 v2 pith:35GOK5LQ submitted 2025-10-20 math.NT

Roth's Theorem in Super Smooth Numbers

Laurence P. Wijaya This is my paper
classification math.NT MSC 11B2511L0711N25
keywords arithmetic progressionsRoth's theoremsmooth numberssuper smooth numbersW-tricktransference principleexponential sumsrestriction estimates
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

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.

Core claim

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

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [§4, application of Proposition 3.6] 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.
  2. [§4, final paragraph: modular vs integer progressions] 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.
  3. [§5, proof of Proposition 4.3] 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.
  4. [§5, proof of Proposition 4.2] 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.
minor comments (5)
  1. [§3.3, Proposition 3.6] In the displayed conclusion, 'f(n+d)f(n+d)' should presumably read 'f(n+d)f(n+2d)'.
  2. [§4, proof of Theorem 1.1] 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.
  3. [§5, proof of Proposition 4.1] 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.
  4. [§5, proof of Proposition 4.2] 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.
  5. [§1, abstract and introduction] 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.

Circularity Check

0 steps flagged

No significant circularity: the derivation rests on external results and contains no fitted-parameter or self-citation chain that forces the conclusion.

full rationale

The proof of Theorem 1.1 is not circular under the specified rubric. The load-bearing inputs are external: Lemma 3.3 and Theorem 3.7 are quoted from Harper [14], Lemma 3.1 from Granville [8], Lemma 3.2 from de la Bretèche–Tenenbaum [3], and Proposition 3.6 from Green–Tao [11]. None of these references states or presupposes the target theorem, and none of the propositions is fitted to the data used in Theorem 1.1. The W-trick decomposes A into fibers A_b and derives lower bounds for the f_b-correlation via the transference principle; no parameter is tuned to force the final 3-AP count, and the conversion from f_b-correlation to 1_{A_b}-correlation is an algebraic identity using the definition of f_b. The serious gap noted by the reader—applying the Green–Tao transference principle, stated for prime N, to composite N_b, and converting modular APs into integer APs—is a correctness/rigor problem, not a circularity problem: it does not reduce the conclusion to an assumption of that conclusion. There is also no self-citation: the sole author cites prior work of others (Harper, Green, Green–Tao) as evidence, and those are independent external results. I therefore find no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No fitted constants or new postulated entities. The ledger lists the imported analytic machinery and the two unverified technical assumptions introduced in the proof: the prime-modulus transference and the applicability of Harper's restriction theorem to A_b.

axioms (4)
  • domain assumption Standard smooth-number distribution lemmas (Lemma 3.1, Lemma 3.2): Ψ(N,y)=N^{1−1/K+o(1)}, uniform distribution in arithmetic progressions for q≤y^β, and Ψ_q(N,y)≈g_q(α)Ψ(N,y).
    §3.1; these external results underpin the W-trick normalization and the count of smooth numbers in the chosen residue class.
  • domain assumption Harper's minor-arc estimate (Lemma 3.3) and restriction theorem (Theorem 3.7) for exponential sums over y-smooth numbers.
    §3.2; used in Propositions 4.2 and 4.3 to control the Fourier transform of the W-tricked measure and the L^p norm of the f_b exponential sum.
  • domain assumption Green–Tao Fourier-analytic transference principle (Proposition 3.6) with N a large prime.
    §3.3; the proof applies it to N_b, which is not shown to be prime, so this is a load-bearing assumption in the submitted form.
  • ad hoc to paper Harper's restriction theorem can be applied to the set A_b = {n : b_1(W n−b_2)∈A} even though A_b is not shown to consist of y-smooth numbers.
    Proof of Proposition 4.3 bounds ∫ |∑_{n∈A_b} e(nθ)|^p via Theorem 3.7 over S(N_b,y) without proving A_b⊆S(N_b,y) and without supplying a residue-class restriction estimate.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Roth's Theorem in Super Smooth Numbers." pith.science (2026). https://pith.science/paper/35GOK5LQ

@misc{pith2026251018024,
  author       = {Pith},
  title        = {Pith review of: Roth's Theorem in Super Smooth Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35GOK5LQ}},
  note         = {Machine review of arXiv:2510.18024}
}
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read the original abstract

We say that the set of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$ is super smooth if $y=\log^KN$ for a large fixed constant $K$. We show that the Roth's theorem on arithmetic progressions is true in super smooth numbers case. This extends the result of Harper where he showed the statement is true under a weaker hypothesis.

discussion (0)

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.