Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions

Abstract

This paper broadens the range on which Fouvry and Radziwi{\l}{\l}'s results on nearly balanced convolutions apply. In particular, let $\alpha_m$ and $\beta_n$ be sequences supported on $m\sim M$ and $n\sim N$ where $\beta_n$ is equidistributed for small moduli, and let $Q=X^{\frac 12+\varepsilon}$. We find that \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*} if $N=X^{\frac 12+\delta}$ and $M=X^{\frac 12-\delta}$ with $0<\delta<\frac 1{68}$, which improves Fouvry and Radziwi{\l}{\l}'s $0<\delta<\frac 1{112}$. To prove this, we sharpen Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where some of the sums are over subdyadic intervals.