Mean value theorems with smooth numbers

F. Majeed; Igor E. Shparlinski

Mean value theorems with smooth numbers

T0 review · 0 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For K>12, the additive energy of smooth numbers saves a power of x over the trivial bound.

desk verdict A solid, honest extension of existing pointwise bounds to new mean value ranges; the main risk is the unverified transcription of Harper's and Baker's bounds, but the internal derivation is clean and the paper merits serious refereeing. read the letter →

arxiv 2506.22192 v1 pith:OUDPOWC4 submitted 2025-06-27 math.NT

F. Majeed , Igor E. Shparlinski This is my paper · ORCID

classification math.NT MSC 11L0711N25

keywords smoothnumbersexponentialsumsmeanvaluetheoremsadditiveenergypowersavingsparsesetsDirichletapproximation

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

The paper proves new mean value bounds for exponential sums over $y$-smooth numbers when $y$ grows like a power of $\log x$, the range where the set is sparse ($\Psi(x,y)=x^{1-\kappa+o(1)}$) but where no previous bound improved on the trivial estimate $\Psi^{\rho-1}$. Its main theorem gives, for admissible $\rho$ and $K$, $I_\rho(x,y) \le \Psi(x,y)^{1+o(1)}(x^{3(\rho-2)/4}+x^{(\rho-2)(1-\eta)})$, with an explicit $\eta$. The corollary for $\rho=4$ is a power saving for the additive energy: for $K>12$, $E(x,y) \le \Psi(x,y)^{3-\zeta+o(1)}$ with $\zeta=\kappa(1-12\kappa)/(1+5\kappa-6\kappa^2)$. This closes part of the gap between the very small $y$ where an optimal bound is known and the larger $y$ where a different pointwise method already works.

What carries the argument

The proof splits $[0,1]$ by Dirichlet approximation into arcs $M_{a,q,Q}$ centered at $a/q$ with $q\le Q$. On arcs with $q\le x^{1/2-\varepsilon}$ it uses a pointwise bound (Lemma 2.2) giving $|S(\theta;x,y)| \le \Psi^{1+o(1)}(qL)^{-\gamma}(\log x)^{5/2+o(1)}$ whenever $qL\le 2x^{1/2-\varepsilon}$; on all arcs it uses an unconditional variant (Lemma 2.3) with an extra $(qL x^{\kappa-1})^{1/2}$ term, where $L=1+x|\theta-a/q|$ and $\gamma=(1-3\kappa)/2$. The large-denominator contribution is controlled by a third pointwise bound (Lemma 2.1). Balancing the two main terms at $Q=x^\xi$ with $\xi=1-2\eta$ converts the pointwise decay into the mean-value exponent $(\rho-2)(1-\eta)$.

What would settle it

A direct check for $K=13$ and increasing $x$ on the major arcs $qL\le 2x^{1/2-\varepsilon}$ would test whether $|S(\theta;x,y)| \le \Psi(x,y)^{1+o(1)}(qL)^{-\gamma}(\log x)^{5/2+o(1)}$ holds; finding a single family of $\theta$ where the right-hand side is exceeded by a power of $\log x$ would invalidate the main saving and force a recomputation of $\zeta$ in Corollary 1.4.

Watch

Extended reading notes

Core claim

Let $S(\theta;x,y)=\sum_{n\in S(x,y)}e^{2\pi i \theta n}$ be the exponential sum over the $y$-smooth numbers up to $x$. The paper establishes that for $y=(\log x)^{K+o(1)}$ with $K>3$, whenever $1/2<\beta\rho<1$ and $2<\rho<2K+4$, the $\rho$-th moment satisfies the bound stated in Theorem 1.3, and that the choice $\rho=4$, $K>12$ yields $E(x,y)=I_4(x,y) \le \Psi(x,y)^{3-\zeta+o(1)}$. Here $\kappa=1/K$, $\beta=(1-3\kappa)/4$, $\eta=\kappa(\rho-1)/(2+3\kappa\rho)$, and $\zeta=\kappa(1-12\kappa)/(1+5\kappa-6\kappa^2)$. Because $\zeta>0$ exactly when $K>12$, the fourth moment is smaller than the trivial $\Psi^3$ by a factor $\Psi^{-\zeta}$, which is a power of $x$ since $\Psi=x^{1-\kappa+o(1)}$. The paper presents this as the first power saving in this intermediate range.

Load-bearing premise

The proof inherits the two pointwise bounds on $|S(\theta;x,y)|$ with exactly the stated exponents, and one of them only under the restriction $qL\le 2x^{1/2-\varepsilon}$; if those quoted bounds are not valid in this form, the saving in the mean values has to be reworked.

Editorial extensions

If this is right

For K>12 the additive energy of the (log x)^K-smooth numbers up to x is bounded by Ψ(x,y)^{3−ζ+o(1)}, a power saving over the trivial Ψ^3.
The same theorem gives nontrivial mean value bounds for every even ρ in the stated range, so the method applies beyond the fourth moment.
The bound on E(x,y) feeds into the known criterion connecting additive energy to the metric Poissonian pair-correlation property, extending the range of K for which smooth numbers have that property.
The argument also yields estimates for the number of solutions to linear equations and for matrix counts with entries from the smooth set, as outlined in the applications section.
The new bounds interpolate between the regime of very small y where an optimal bound is known and the regime of larger y where a different method already gave power savings.

Reading between the lines

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

A sharper version of the quoted pointwise bounds could push the threshold in Corollary 1.4 from K>12 toward K>4, the natural boundary where the trivial bound currently takes over; testing this would require revisiting the two imported lemmas rather than the arc-splitting argument itself.
The exponent ζ has the form κ(1−12κ)/(1+5κ−6κ^2), and a matching lower bound is not addressed: one could try to construct many distinct solutions to n1+n2=n3+n4 with all ni y-smooth to see whether the energy is forced to stay close to Ψ^3 for K just above 12.
Applying the same balancing at Q=x^{1/2} exactly, rather than with the ε-slack used here, might replace the o(1) exponent by a computable logarithmic factor and make the bound directly comparable with numerical data for moderate x.
The method could be extended to weighted smooth numbers or to products of smooth numbers; the arc-splitting would survive, but the pointwise inputs would need to be re-verified.

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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

Read this one if you care about mean values of exponential sums over y-smooth numbers in the intermediate range y = (log x)^K. The paper does exactly what it says: it takes the simplified pointwise bounds of Harper, Baker, and Fouvry–Tenenbaum, runs them through a Dirichlet partition, and produces mean value theorems that are nontrivial in ranges not covered by the earlier small-y or large-y results. The most concrete payoff is Corollary 1.4, a power-of-x saving for the additive energy of y-smooth numbers when K > 12. That corollary is new and genuinely follows from the main theorem.

What I like: the paper is transparent about its structure. It states the nontriviality conditions explicitly, the balancing of terms in Sections 3–4 is internally consistent, and the authors do not oversell. They flag that the pointwise bounds are simplified from the literature and they do not claim to have re-proved them. The citation pattern is appropriate, with no fitted parameters and no circularity. The internal algebra checks out; I followed the derivation with beta = (1 – 3κ)/4 and gamma = 2β, and the thresholds in (1.7) and (1.8) line up.

The soft spot is exactly the one the stress-test note identifies: the entire saving exponent zeta in Corollary 1.4 depends on the precise form of Lemma 2.2 (Harper) and Lemma 2.3 (Baker), especially the exponent of (qL) and the condition (2.2). If Baker's theorem actually carries an additional restriction on qL, or if his second term has a different x-power, then the H7/H5 bounds and the balancing exponent xi in (4.7) shift, and the stated power saving could shrink or vanish. This is not a flaw in the paper's derivation, because the lemmas are quoted from published work and the authors are upfront about simplification. But it is load-bearing, and a referee should check those transcriptions against the originals before trusting Corollary 1.4 quantitatively. The paper would not collapse if a minor exponent changed—the general method would survive—but the exact statement might need adjustment.

Minor issues: a couple of typos and a duplicated sentence in Section 1.1, but nothing that obscures the mathematics.

Who is this for? Number theorists working on smooth numbers, mean values, and additive energy. It is a modest but real advance, not a breakthrough. I would send it to a competent referee rather than desk-reject it, and I would expect the referee's main job to be verification of the quoted external bounds.

Recommendation: accept after a check of Lemmas 2.2 and 2.3 against Harper's and Baker's theorems. The paper deserves referee time.

Referee Report

0 major / 5 minor

Summary. The paper proves two mean value theorems for exponential sums over y-smooth numbers up to x in the range y=(log x)^{K+o(1)}. Theorem 1.1 gives a bound for I_rho(x,y) when K>3 and beta rho>1/2, with a nontrivial range rho>4(K-1)/(K-3), K>4. Theorem 1.3 targets the additive energy case and, after balancing major-arc and minor-arc contributions, yields Corollary 1.4: for K>12, E(x,y) <= Psi(x,y)^{3-zeta+o(1)} with zeta = kappa(1-12kappa)/(1+5kappa-6kappa^2) and kappa=1/K. The proofs use Dirichlet approximation, the Parseval identity, and pointwise bounds quoted from Fouvry-Tenenbaum (Lemma 2.1), Harper (Lemma 2.2), and Baker (Lemma 2.3).

Significance. If correct, the paper supplies the first power savings for the additive energy of (log x)^K-smooth numbers in the intermediate range K>12, interpolating between the small-y results of Bourgain-Garaev-Konyagin-Shparlinski and the large-y mean value theorem of Harper. The internal algebra of Sections 3 and 4 is consistent, there are no fitted parameters, and the dependence on the quoted external pointwise bounds is explicit. The main caveat is that the final saving exponent zeta is directly computed from the simplified forms of Harper's and Baker's theorems in Lemmas 2.2 and 2.3; I found no demonstrated misquotation, but the manuscript would be easier to certify if those forms were located precisely in the cited sources. The paper is clearly written and the claimed result is concrete and falsifiable.

minor comments (5)

[Section 4.1] The first sentence says 'We proceed as in the proof of Theorem 1.3', but the surrounding argument is a variant of the proof of Theorem 1.1; this should be corrected.
[Section 4.3] The phrase 'we now chose Q' should read 'we now choose Q'.
[Section 1.1] The sentence 'which follows from the Parseval identity I2(x,y)=Psi(x,y)' is duplicated verbatim in the displayed text; the duplicate should be removed.
[Sections 3.3 and 4.2] The step replacing the sum over Farey arcs by I2(x,y), namely 'sum_{q,a} int_{M_{a,q,Q}} |S|^2 dtheta ! I2(x,y)', should be justified, because the arcs M_{a,q,Q} overlap. With the standard bounded-overlap (up to O(log Q)) property this is harmless and is absorbed by the x^{o(1)} factor, but the justification should be stated explicitly.
[Section 2.2, Lemmas 2.2 and 2.3] Since the exponent gamma=(1-3kappa)/2 and the second term (qL x^{kappa-1})^{1/2} in Lemma 2.3 directly determine the saving exponent zeta in Corollary 1.4, the authors should add a short note confirming that these simplified forms follow from [18, Theorem 1] and [2, Theorem 2] with no additional hidden restrictions on qL or on the range of theta. This would remove a genuine verification burden on the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main mean value bounds are derived from external pointwise bounds (Harper, Baker, Fouvry–Tenenbaum) and standard smooth-number estimates; the authors' own cited results are contextual, not load-bearing.

full rationale

The derivation of Theorems 1.1 and 1.3 starts from Dirichlet approximation (3.1) and then applies Lemma 2.1 (Fouvry–Tenenbaum), Lemma 2.2 (Harper), and Lemma 2.3 (Baker) as external pointwise bounds on S(ϑ;x,y). No parameter is fitted to the mean values being bounded, and the final saving exponents arise by explicit balancing: for Theorem 1.3, substituting the estimates (4.2) and (4.6) gives the balancing equation (4.7), whose solution ξ = 1 − 2η determines the exponent ζ in Corollary 1.4 algebraically. The only self-citation that appears in the relevant range is [6], used in the introduction for the known bound (1.5) on I_{2s}(x,y), but (1.5) is not invoked in the proofs of Theorem 1.1 or Theorem 1.3; those proofs rely on the external lemmas and on (1.2). The paper's use of 'simplified forms' of Harper's and Baker's theorems is a faithfulness-of-citation concern, not a circularity concern: the quoted lemmas have assumptions independent of the conclusions, and the paper computes consequences rather than assuming them. Hence no circular step, self-definitional reduction, or fitted-input-as-prediction is present.

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

There are no free parameters and no invented entities. The constants kappa, beta, gamma, zeta and eta are derived functions of the fixed input parameters K and rho, not fitted to data. All substantive analytic inputs are quoted pointwise bounds from the published literature (Harper, Baker, Fouvry-Tenenbaum) plus the standard smooth-number counting estimate; the paper itself supplies the summation and optimization arguments.

assumptions (5)

domain assumption Psi(x,(log x)^K) = x^(1-kappa+o(1)) for fixed K>1, with kappa=1/K.
Equation (1.2), cited to Granville and Hildebrand-Tenenbaum; used throughout to convert Psi^a into x^(a(1-kappa+o(1))).
domain assumption Harper's pointwise bound, Lemma 2.2: for qL <= 2x^(1/2-epsilon), |S| <= Psi^(1+o(1)) (qL)^(-gamma+o(1)) (log x)^(5/2+o(1)) with gamma=2beta=(1-3kappa)/2.
Quoted in simplified form from Harper [18]; it is the main saving mechanism in the proof of Theorem 1.1 and in the small-q part of Theorem 1.3.
domain assumption Baker's variant, Lemma 2.3: unconditionally, |S| <= Psi^(1+o(1))((qL)^(-gamma) + (qL x^(kappa-1))^(1/2)).
Quoted from Baker [2]; it removes Harper's restriction qL <= 2x^(1/2-epsilon) at the cost of an extra term, and is used in Section 4.2.
domain assumption Fouvry-Tenenbaum bound, Lemma 2.1: |S| <= x^(1+o(1))(x^(-1/4)+q^(-1/2)+(q/x)^(1/2)) L.
Quoted from Fouvry-Tenenbaum [15]; used for the large-q ranges I2 and J2, where Harper's saving is no longer available.
standard math Parseval identity I2(x,y)=Psi(x,y) and Dirichlet approximation covering of [0,1] by the intervals M_{a,q,Q}.
These are standard tools invoked in Section 3.1; the Parseval identity is the baseline for the trivial bound, and the Dirichlet partition is the starting splitting of the mean value integral.

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Cite this review

Pith. "Pith review of Mean value theorems with smooth numbers." pith.science (2026). https://pith.science/paper/OUDPOWC4

@misc{pith2026250622192,
  author       = {Pith},
  title        = {Pith review of: Mean value theorems with smooth numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUDPOWC4}},
  note         = {Machine review of arXiv:2506.22192}
}

read the original abstract

We obtain new mean value theorems for exponential sums with very smooth numbers, which provide a power saving against the trivial bound in region where previous bounds do not apply.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Aistleitner, G

C. Aistleitner, G. Larcher and M. Lewko, ‘Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems’, Israel J. Math., 222 (2017), 463–485. 2, 5

work page 2017

[2] [2]

R. C. Baker, ‘Smooth numbers in Beatty sequences, Acta Arith. 200 (2021), 429–438. 7

work page 2021

[3] [3]

R. C. Baker, M. Munsch and I. E. Shparlinski, ‘Additive energy and a large sieve inequality for sparse sequences’, Mathematika 68 (2022), 362–399. 5

work page 2022

[4] [4]

Blomer and J

V. Blomer and J. Li, ‘Correlations of values of random diagonal forms’, Intern. Math. Res. Notices 2023 (2023), 20296–20336. 6 MEAN V ALUE THEOREMS WITH SMOOTH NUMBERS 15

work page 2023

[5] [5]

T. F. Bloom and A. Walker, ‘GCD sums and sum-product estimates’, Israel J. Math. 235 (2020), 1–11. 2, 5

work page 2020

[6] [6]

Bourgain, M

J. Bourgain, M. Z. Garaev, S. V. Konyagin and I. E. Shparlinski, ‘Multiplica- tive congruences with variables from short intervals’, J. d’Analyse Math. 124 (2014), 117–147. 3

work page 2014

[7] [7]

Br¨ udern and T

J. Br¨ udern and T. D Wooley,Estimates for smooth Weyl sums on major arcs , Intern. Math. Res. Not 2024 (2024), 14662–14688. 7

work page 2024

[8] [8]

N. G. de Bruijn, ‘On the number of positive integers ď x and free of prime factors ą y, II’, Indag. Math. 28 (1966), 239–247. 3

work page 1966