KLOOSTERMAN SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS

We bound Kloosterman-like sums of the shape Sigma(N)(n=1) exp(2 pi i(x[f(n) + y[y(n)](-1))/p), with integers parts of a real-valued, twice-differentiable function f is satisfying a certain limit condition on f'', and [f(n)](-1) is meaning inversion modulo p. As an immediate application, we obtain results concerning the distribution of modular inverses inverses [f(n)](-1) (mod p). The results apply, in particular, to Piatetski-Shapiro sequences [t(c)] with c is an element of (1, 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.