Bounds of triple exponential sums with mixed exponential and linear terms

TL;DR

This paper derives bounds for complex triple exponential sums involving mixed exponential and linear functions, utilizing advanced combinatorial and number-theoretic methods to improve understanding of their behavior.

Contribution

It introduces new bounds for triple exponential sums with mixed terms, combining techniques from Shparlinski and recent additive energy bounds from Roche-Newton, Rudnev, and Shkredov.

Findings

Established new bounds for triple exponential sums with mixed functions.

Applied combined methods from different researchers to improve sum estimates.

Enhanced understanding of exponential sum behavior in number theory.

Abstract

We establish bounds of triple exponential sums with mixed exponential and linear function. The method we use is by Shparlinski together with a bound of additive energy from Roche-Newton, Rudnev and Shkredov.