Kloosterman sums with twice-differentiable functions

Abstract

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2πi (x \lfloor f(n)\rfloor+ y \lfloor f(n)\rfloor^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f^{\prime \prime }$, and $\lfloor f(n){\rfloor }^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\lfloor f(n){\rfloor }^{-1}(\mathrm{mod}p)$. The results apply, in particular, to Piatetski-Shapiro sequences $\lfloor t^c\rfloor$ 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.