Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced convolutions

Abstract

This paper broadens the range on which Fouvry and Radziwiłł'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 $Q=X^{\frac{1}{2}+\epsilon }$. 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}}α_mβ_n-\frac{1}{ϕ(q)}\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ (mn,q)=1}}α_mβ_n\right|\ll \frac{X}{\log^A X} \end{gather*} if $N=X^{\frac{1}{2}+\delta }$ and $M=X^{\frac{1}{2}-\delta }$ with $0<\delta <\frac{1}{68}$, which improves Fouvry and Radziwiłł'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.