# Kloosterman sums with twice-differentiable functions

## TL;DR

This paper establishes bounds on Kloosterman-like sums involving twice-differentiable functions with specific curvature conditions, leading to new insights into the distribution of modular inverses of sequences like Piatetski-Shapiro sequences.

## Contribution

It introduces novel bounds for sums involving the floor of twice-differentiable functions and applies these to analyze the distribution of modular inverses in sequences such as Piatetski-Shapiro sequences.

## Key findings

- Bounds on Kloosterman-like sums for specific functions
- Results on the distribution of modular inverses in certain sequences
- Application to Piatetski-Shapiro sequences with c in (1, 4/3)

## Abstract

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