Consecutive Quadratic Residues and Primitive Roots in the Sequences Formed by Twice-differentiable Functions

Abstract

In this paper we bound character sums of the shape \[ \sum_{n \leq N} \chi_1(\lfloor f(n) \rfloor) \chi_2(\lfloor f(n+l) \rfloor), \] where $\chi_1$ and $\chi_2$ are non-principal multiplicative characters modulo a prime $p$, $f(x)$ is a real-valued, twice-differentiable function satisfying a certain condition on $f''(x)$, and $l$ is a positive integer. As an immediate application, we obtain some distribution properties of consecutive quadratic residues and consecutive primitive roots in Piatetski–Shapiro sequences $\lfloor n^c \rfloor$ with $c \in (1,4/3)$.