Kloosterman sums with twice-differentiable functions

Abstract

We bound Kloosterman-like sums of the shape N ÿ n"1 expp2πipxtf pnqu`ytf pnqu´1q{pq, with integers parts of a real-valued, twice-differentiable function f is satisfying a certain limit condition on f 2 , and tf pnqu´1 is meaning inversion modulo p.As an immediate application, we obtain results concerning the distribution of modular inverses inverses tf pnqu´1 pmod pq. The results apply, in particular, to Piatetski-Shapiro sequences tt c u with c P p1, 4 3 q. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from