AI 中文总结
该论文拓展了Fouvry与Radziwiłł近平衡卷积结果的适用范围，将参数δ的上限从1/112提升至1/68，其关键在于改进了含Kloosterman分数的三重型在次二进区间求和情形下的结论。
AI 中文摘要
本文拓展了Fouvry与Radziwiłł关于近平衡卷积结果的适用范围。具体而言，设αₘ与βₙ为支撑在m~M和n~N上的序列，其中βₙ对小模长均匀分布，且Q=X^(1/2+ε)。当N=X^(1/2+δ)、M=X^(1/2−δ)且0<δ<1/68时，我们证明∑_{q~Q}|∑_{n~N,m~M, mn≡a mod q}αₘβₙ − 1/φ(q)∑_{n~N,m~M, (mn,q)=1}αₘβₙ| ≪ X/log^A X，此结果改进了Fouvry与Radziwiłł的0<δ<1/112的结论。为证明该结论，我们在部分求和针对次二进区间的情形下，改进了Bettin与Chandee关于含Kloosterman分数的三重型的著名结果。
英文摘要
This paper broadens the range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply. In particular, let $α_m$ and $β_n$ be sequences supported on $m\sim M$ and $n\sim N$ where $β_n$ 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}}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 $N=X^{\frac 12+δ}$ and $M=X^{\frac 12-δ}$ with $0<δ<\frac 1{68}$, which improves Fouvry and Radziwiłł's $0<δ<\frac 1{112}$. 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.
