Trilinear forms with Kloosterman fractions

Abstract

We give new bounds for ${\sum }_{a,m,n}{\alpha }_m{\beta }_n{\nu }_a\text{e}\left(\frac{a\overline{m}}{n}\right)$ 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.