Kloosterman sums with twice-differentiable functions

Abstract

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \floor{f(n)}+ y \floor{f(n)}^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\floor{f(n)}^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\floor{f(n)}^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \floor{t^c}$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.