Trilinear forms with Kloosterman fractions

TL;DR

This paper establishes improved bounds for certain exponential sums involving Kloosterman fractions and applies these results to problems in number theory such as determinant equations and solution distribution.

Contribution

It introduces new bounds for exponential sums with Kloosterman fractions, enhancing previous results by Duke, Friedlander, and Iwaniec, and applies these bounds to various number-theoretic problems.

Findings

Improved bounds for exponential sums with Kloosterman fractions.

Applications to representation problems involving determinant equations.

Results on the equidistribution of solutions to linear equations.

Abstract

We give new bounds for $\sum_{{a, m ,n}}\alpha_{m}\beta_n\nu_a {\textrm e}\left(\frac{a\overline m}{n}\right)$ where $\alpha_{m}$, $\beta_n$ and $\nu_a$ are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec [DFI97]. We also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations.