Exponential sums with Piatetski-Shapiro sequences and their application to modular hyperbolas

Abstract

The Piatetski-Shapiro sequence associated with c>1 and c∉N is given by Nc=(⌊nc⌋)n∈N. Let p be an odd prime and let g(x),h(x)∈Fp[x] be coprime polynomials. We estimate exponential sums with Piatetski-Shapiro sequences of the shape∑m≤Mm∈Nc,(h(m),p)=1ep(g(m)h(m)), where ep(y)=exp⁡(2πiy/p). In particular, when g(x)/h(x)=rx+sx¯ and 1<c<6/5, our results improve the bound for Kloosterman sums with Piatetski-Shapiro sequences which was given by Shparlinski and Technau. Furthermore, bounds for max⁡{m,m˜} subject to m,m˜∈Z∩[1,p), z indivisible by p, g(m)m˜≡z(modp) and m belonging to some fixed Piatetski-Shapiro sequence are also obtained.