We first prove an extension of the Bourgain–Sarnak–Ziegler theorem, relaxing some conditions and giving quantitative estimates. Then we apply our extension to bound certain exponential sums, where the coefficients come from modular forms and the exponential involves polynomial sequences of any degree.

中文翻译：

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BOURGAIN-SARNAK-ZIEGLER定理在模块应用中的扩展

我们首先证明Bourgain–Sarnak–Ziegler定理的扩展，放宽某些条件并给出定量估计。然后，我们将扩展应用于有界的某些指数和，其中系数来自模数形式，并且指数涉及任何程度的多项式序列。