Incomplete exponential sums over exponential functions

TL;DR

This paper extends methods for bounding exponential sums involving exponential functions to cases where the base is not a primitive root, and connects recent results on additive properties of subgroups to new bounds.

Contribution

It introduces new bounds for exponential sums when the base is not a primitive root, leveraging additive properties of multiplicative subgroups.

Findings

Extended bounds for exponential sums with non-primitive roots

Linked subgroup additive properties to sum bounds

Provided new theoretical bounds in number theory

Abstract

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2\pi iag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.