Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

TL;DR

This paper advances bounds on unbalanced convolutions involving Kloosterman fractions by refining existing results and extending the applicable ranges of parameters, using improved estimates for trilinear forms.

Contribution

It introduces new bounds for unbalanced convolutions with partially fixed moduli, extending the parameter ranges for N and Q, based on improved estimates of Kloosterman fractions.

Findings

Extended the range of N for which bounds hold.

Widened the bounds for Q in unbalanced convolutions.

Improved estimates for trilinear forms with fixed denominator factors.

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.