bibkey: bettin2018trilinear
authors: Sandro Bettin; Vorrapan Chandee
year: 2018
title: "Trilinear forms with Kloosterman fractions"
doi: 10.1016/j.aim.2018.01.026
url: https://arxiv.org/abs/1502.00769v1
claim: "Theorem 1 bounds trilinear forms with inverse-modulus phases; its single-third-variable specialization has no power saving at the Robin residual's 3:1 candidate alias split, even before the missing phase conversion. The stronger all-aspect-ratio claim in arXiv:2601.00292v1 is withdrawn and is not an input."
strata_touched: []
license: citation-only
triage: anchor

Kloosterman-fraction supplier and its applicability boundary

The inspected mathematical text is Theorem 1, equation (1.2), in arXiv:1502.00769v1, submitted 3 February 2015. The arXiv version record lists that single preprint version and the journal reference Advances in Mathematics 328 (2018), 1234–1262. The DOI above identifies the journal publication; this note does not assert a comparison of its proof with the preprint. The bounds below are cited or directly specialized, not reproved or claimed as new mathematics. No Lean verification is supplied.

The source contract

Write $e(u)=\exp (2\pi iu)$ and let $\Vert \cdot {\Vert }_2$ denote the norm of a coefficient sequence on its stated support. For positive lengths $M_1,M_2,U$, take arbitrary complex coefficients supported on $[M_1/2,M_1]$, $[M_2/2,M_2]$, $[U/2,U]$, respectively. Restricting to a fixed nonzero integer $\vartheta$ suffices here. Put $$\mathcal{B}(M_1,M_2,U)=\sum _{\begin{array}{c}a\in [U/2,U],m\in [M_1/2,M_1],n\in [M_2/2,M_2]\\ (m,n)=1\end{array}}{\alpha }_m{\beta }_n{\nu }_{a}e\left(\vartheta \frac{a\overline{m}}{n}\right),m\overline{m}\equiv 1(\mathrm{mod}n).$$ For every fixed $\eta >0$, Theorem 1 supplies $$\begin{array}{rl}|\mathcal{B}(M_1,M_2,U)|{\ll }_{\eta }& \Vert \alpha {\Vert }_2\Vert \beta {\Vert }_2\Vert \nu {\Vert }_2{\left(1+\frac{|\vartheta |U}{M_1{M}_2}\right)}^{1/2}\\ & \times \left[(U{M}_1{M}_2{)}^{7/20+\eta }(M_1+M_2{)}^{1/4}+(U{M}_1{M}_2{)}^{3/8+\eta }(U{M}_1+U{M}_2{)}^{1/8}\right].\end{array}\text{\hspace{1em}(BC1)}$$ The two terms are added. The theorem does not allow choosing their minimum. The modular inverse and the coprimality condition belong to the source’s object; an additive phase $e(mr/q)$ is a different input.

Direct single-variable specialization

Set $U=1$ and ${\nu }_1=1$, so the third variable has one value. Let $X\to \infty$, and suppose $$M_1=X^{t+o(1)},M_2=X^{1-t+o(1)},\frac{1}{2}\le t<1.$$ With $\vartheta$ fixed, the factor in square brackets in (BC1) is $$X^{7/20+t/4+\eta +o(1)}+X^{3/8+t/8+\eta +o(1)}.\text{\hspace{1em}(BC2)}$$ Both have the same coefficient-norm prefactor. The elementary Cauchy–Schwarz bound has instead the factor $(M_1{M}_2{)}^{1/2}=X^{1/2+o(1)}$. A power saving from (BC2), for a sufficiently small fixed $\eta$, requires $t<3/5$; in this range both exponents without $\eta$ are strictly below $1/2$. At $t=3/4$, they are respectively $$\frac{43}{80}>\frac{1}{2},\frac{15}{32}<\frac{1}{2}.\text{\hspace{1em}(BC3)}$$ Consequently (BC1) does not supply a power saving for this split. Taking the minimum with the independent elementary bound remains available, but supplies no power saving either. This is a comparison of upper-bound allowances, not a lower bound on the actual form.

Interface to the retained Robin residual

The owning residual application, equations (I18)–(I19) and (I105)–(I106), retains the actual aliases $$b_{N,q}=\sum _{j\le N/q}\frac{a_{N,qj}}{j}$$ and the phases $e(mr/q)$, with every reduced numerator and their original weights. A candidate factorization near $q=N^{3/4+o(1)}$ and $qj\asymp N$ has lengths $N^{3/4+o(1)}:N^{1/4+o(1)}$. Even if a conversion to (BC1) with a single third variable and no additional losses were available, (BC3) would give no power saving. No such phase and weight conversion is established here.

Moreover $q>N/2$ has only the alias $j=1$. A bilinear label does not furnish a second long averaging interval in that sector. The source does not pay the constant $d_N$, the actual endpoint, or their interaction with the retained middle frequencies.

This excludes this direct specialization as a quantitative supplier. It does not exclude a different factorization, a genuine third average, a valid spectral transformation or additional structure in the actual coefficients. None of those interfaces is supplied by renaming the original sum.

Withdrawn stronger claim

The current arXiv record for 2601.00292 marks Dong–Robles–Zeindler, Bilinear forms with Kloosterman fractions and applications, as withdrawn in v2, 5 January 2026. Its withdrawal comment states that a missed factor $L^2$ in equation (2.53) changes $L^5$ to $L^7$, and the argument does not give the claimed improvement. Here $L$ is that paper’s amplification parameter.

The superseded v1 contains the stronger all-aspect-ratio bound in Theorem 1.6. That bound is not an input to this note or to the Robin application. The withdrawal and its stated reason are existing source information, not a new error discovery or a new obstruction theorem. No claim about every other result in that withdrawn manuscript is needed.