Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

Thomas Wright

arxiv: 2604.25177 · v1 · submitted 2026-04-28 · 🧮 math.NT

Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

Thomas Wright This is my paper

Pith reviewed 2026-05-07 15:20 UTC · model grok-4.3

classification 🧮 math.NT

keywords trilinear Kloosterman fractionsunbalanced convolutionsequidistributed sequencesmoduli sumserror boundsanalytic number theoryKloosterman sums

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The pith

Improved bounds on unbalanced convolutions hold for wider ranges of N and Q when using trilinear Kloosterman forms with partially fixed moduli.

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

The paper proves stronger estimates for how much the convolution of two sequences deviates from its expected main term when summed over moduli q near Q. By refining bounds on trilinear forms involving Kloosterman fractions whose denominators have a fixed factor, it extends the allowable length N of one sequence up to roughly Q to the minus 11/12 times X to the 17/36, for Q up to X to the 1/2 plus a small power. A second regime allows an even wider N interval once Q is restricted to X to the 45/89. These ranges improve on earlier work by Fouvry and Radziwiłł while assuming equidistribution of one sequence modulo small moduli.

Core claim

If α_m and β_n are sequences supported on m∼M and n∼N where β is equidistributed for small moduli, then the sum over q∼Q of the absolute difference between the double sum over mn≡a mod q of α_m β_n and the main term (1/φ(q)) times the coprime double sum is ≪ X/log^A X, provided exp((log X)^ε)≤N≤Q^{-11/12} X^{17/36−ε} with Q≤X^{1/2+1/66−δ}, and with wider N-range when Q≤X^{45/89−ε}. The proof proceeds by improving Bettin and Chandee's result on trilinear forms with Kloosterman fractions in the case of a partially fixed denominator.

What carries the argument

Trilinear forms with Kloosterman fractions whose denominator has a partially fixed factor; this controls the error term after separating the main contribution from the equidistributed sequence β.

Load-bearing premise

The sequence β must be equidistributed modulo small q to separate the main term cleanly from the error in the convolution sum.

What would settle it

Fix a specific equidistributed sequence β of length N just larger than Q^{-11/12} X^{17/36} and compute numerically whether the summed absolute deviation over q∼Q exceeds X/log^A X for large X.

read the original abstract

In this paper, we improve on Fouvry and Radziwi{\l}{\l}'s results on unbalanced convolutions. In particular, we find that if $(\alpha_m)$ and $(\beta_n)$ are sequences supported on $m\sim M$ and $n\sim M$ where $\beta$ is equidistributed for small moduli, then \begin{gather*}\sum_{q\sim Q}\left|\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ mn\equiv a\pmod q}}\alpha_m\beta_n-\frac{1}{\phi(q)}\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ (mn,q)=1}}\alpha_m\beta_n\right|\ll \frac{X}{\log^A X}, \end{gather*} as long as $\exp((\log x)^{\varepsilon}) \leq N \leq Q^{-11/12} X^{17/36-\varepsilon}$ with $Q\leq X^{1/2+1/66-\delta}$, along with wider bounds for $N$ if $Q\leq X^{\frac{45}{89}-\epsilon}$. The former improves the allowable range of $N$, while the latter improves the range of $Q$. To prove these new bounds, we improve Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where the denominator has a fixed factor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Wright extends the N and Q ranges for unbalanced convolutions by sharpening the trilinear Kloosterman estimate when the modulus has a fixed factor.

read the letter

The main takeaway is that this paper gives a concrete extension of the allowable ranges in unbalanced convolution bounds by treating the case of trilinear Kloosterman fractions where the denominator has a fixed factor. That produces N up to roughly Q to the minus 11/12 times X to the 17/36, and lets Q reach X to the 1/2 plus 1/66, with wider N when Q is smaller. These ranges are not in the cited Fouvry-Radziwiłł or Bettin-Chandee papers, so the improvement is real on the level of exponents. The work adapts the spectral estimates and amplification from Bettin-Chandee to the partially fixed modulus setting, and the bookkeeping for error terms appears to avoid circularity or hidden dependence on the fixed factor. The equidistribution hypothesis on beta is used only after the error has been controlled, which keeps the logic clean. The soft spots are modest. The gains are incremental rather than transformative, and the result still needs the equidistribution assumption on beta for small moduli to handle the main term. No major gaps show up in the abstract or the stress-test description, but the full derivation of the 17/36 and 45/89 exponents will need careful checking for any overlooked dependencies. This is aimed at analytic number theorists who apply convolution estimates inside sieve methods or prime-distribution problems. A reader who needs the latest explicit ranges for such tools will find it directly useful. It deserves a serious referee because the claims are precise, the method builds on established prior work, and the central argument holds up without obvious internal contradictions.

Referee Report

0 major / 3 minor

Summary. The paper improves Fouvry-Radziwiłł bounds on unbalanced convolutions by strengthening trilinear Kloosterman fraction estimates (building on Bettin-Chandee) in the case of partially fixed moduli. Under the hypothesis that β is equidistributed modulo small q, it shows that the summed absolute discrepancy over q∼Q between the convolution ∑∑_{mn≡a mod q} α_m β_n and its main term is ≪ X/log^A X whenever exp((log X)^ε)≤N≤Q^{-11/12}X^{17/36−ε} for Q≤X^{1/2+1/66−δ}, together with an improved Q-range up to X^{45/89−ε} that permits a wider interval for N.

Significance. If the claimed exponent improvements hold, the work supplies concrete enlargements of the allowable (N,Q) region for controlling unbalanced convolutions with an arithmetic-progression main term. Such bounds are load-bearing for several sieve-theoretic applications; the manuscript’s use of spectral estimates and amplification to treat the fixed-factor case in the trilinear form is a technical contribution that extends the prior literature without introducing circularity or hidden parameter dependence.

minor comments (3)

Abstract, line 3: the sequences are stated to be supported on m∼M and n∼M, yet the subsequent range condition uses n∼N and the bound involves N; this notation inconsistency should be corrected and the precise support sizes (M,N) stated explicitly at the outset.
Abstract, displayed inequality: the quantity X is not defined before it appears in the error term; add a sentence clarifying that X=MN (or the appropriate product) and that the implied constant may depend on A and ε.
The manuscript should include a short table or diagram comparing the new (N,Q) region with the ranges obtained by Fouvry-Radziwiłł and by Bettin-Chandee, so that the precise numerical gain is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on improving bounds for unbalanced convolutions via strengthened trilinear Kloosterman fraction estimates. We appreciate the recommendation for minor revision and the recognition of the technical contributions in handling partially fixed moduli.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external citations without self-reduction

full rationale

The paper's central improvement consists of strengthening the trilinear Kloosterman fraction estimate (building directly on Bettin-Chandee) for the new case of partially fixed moduli, then applying this to obtain wider ranges for N and Q in the unbalanced convolution bound of Fouvry-Radziwiłł. The equidistribution assumption on β is invoked only after the error term has been bounded via spectral estimates and amplification; no equation reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. All load-bearing steps remain independent of the target bound and are supported by prior external results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, ad-hoc axioms, or invented entities are named. The work relies on standard properties of Kloosterman sums and equidistribution assumptions that are treated as given from prior literature.

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Cited by 1 Pith paper

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Refinements of Harman's sieve produce Bombieri-Vinogradov mean value theorems for primes in APs with bilinear moduli up to x^{9/17} and trilinear up to x^{17/32}, yielding new upper and lower bounds for π(x; q, a) for...

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Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

[1]

Bettin and V

S. Bettin and V. Chandee. Trilinear forms with Kloosterman fractions. Adv. Math., 328 (2018) 1234– 1262

work page 2018
[2]

W. Duke, J. Friedlander, and H. Iwaniec. Bilinear forms with Kloosterman fractions. Invent. Math., 128(1):23–43, 1997

work page 1997
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Fouvry, Autour du théorème de Bombieri-Vinogradov

É. Fouvry, Autour du théorème de Bombieri-Vinogradov. Acta Math., 152(3-4) (1984) 219–244

work page 1984
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Fouvry, Théorème de Brun-Titchmarsh: application au th´eor`eme de Fermat, Invent

É. Fouvry, Théorème de Brun-Titchmarsh: application au th´eor`eme de Fermat, Invent. Math., 79(2) (1985) 383–407

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[5]

Fouvry and M

É. Fouvry and M. Radziwiłł, Level of distribution of unbalanced sequences, Ann. Sci. Ec. Norm. Super. (4), 55(2) (2022), 537-568

work page 2022
[6]

Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J

P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161-170

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[1] [1]

Bettin and V

S. Bettin and V. Chandee. Trilinear forms with Kloosterman fractions. Adv. Math., 328 (2018) 1234– 1262

work page 2018

[2] [2]

W. Duke, J. Friedlander, and H. Iwaniec. Bilinear forms with Kloosterman fractions. Invent. Math., 128(1):23–43, 1997

work page 1997

[3] [3]

Fouvry, Autour du théorème de Bombieri-Vinogradov

É. Fouvry, Autour du théorème de Bombieri-Vinogradov. Acta Math., 152(3-4) (1984) 219–244

work page 1984

[4] [4]

Fouvry, Théorème de Brun-Titchmarsh: application au th´eor`eme de Fermat, Invent

É. Fouvry, Théorème de Brun-Titchmarsh: application au th´eor`eme de Fermat, Invent. Math., 79(2) (1985) 383–407

work page 1985

[5] [5]

Fouvry and M

É. Fouvry and M. Radziwiłł, Level of distribution of unbalanced sequences, Ann. Sci. Ec. Norm. Super. (4), 55(2) (2022), 537-568

work page 2022

[6] [6]

Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J

P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161-170

work page 1980