Kloosterman sums with twice-differentiable functions

We bound Kloosterman-like sums of the shape $$\sum _{n=1}^N\exp (2\pi i(x\lfloor f(n)\rfloor +y\lfloor f(n){\rfloor }^{-1})\mathrm{/}p),$$ with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f^{\mathrm{\prime }\mathrm{\prime }}$, 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 $\lfloor f(n){\rfloor }^{-1}(\mathrm{m}\mathrm{o}\mathrm{d}p)$. 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.