# An extension of the Bourgain-Sarnak-Ziegler theorem with modular applications

## TL;DR

This paper extends a key theorem in number theory and applies it to effectively bound polynomial exponential sums with modular coefficients, advancing understanding in analytic number theory.

## Contribution

It introduces an extension of the Bourgain-Sarnak-Ziegler theorem and demonstrates its application to bounding exponential sums with modular coefficients.

## Key findings

- Extended the Bourgain-Sarnak-Ziegler theorem
- Bounded polynomial exponential sums with modular coefficients
- Enhanced tools for analytic number theory

## Abstract

We prove an extension of the Bourgain-Sarnak-Ziegler theorem and then apply it to bound certain polynomial exponential sums with modular coefficients.