Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions

In this paper, we improve on Fouvry and Radziwiłł’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}}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*} as long as $\exp ((\log x{)}^{\epsilon })\le N\le Q^{-11\mathrm{/}12}{X}^{17\mathrm{/}36-\epsilon }$ with $Q\le X^{1\mathrm{/}2+1\mathrm{/}66-\delta }$, along with wider bounds for $N$ if $Q\le X^{\frac{45}{89}-\epsilon }$. 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.