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

In this paper we bound character sums of the shape Sigma(n <= N) chi(1)(left perpendicularf(n)right perpendicular)chi(2)(left perpendicularf(n + l)right perpendicular), 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 left perpendicularn(c)right perpendicular with c is an element of (1, 4/3).