Short exponential sums with a non-integer power of a natural number

Abstract

The following nontrivial estimate is obtained for short exponential sums: $$Sc\left( {\alpha ,x,y} \right) = \sum\limits_{x - y < n \leqslant x} {e\left( {\alpha \left[ {n^c } \right]} \right) < < y\ln ^A x,}$$ where $$y \geqslant x^{\tfrac{1} {2}} \ln ^A x,x^{1 - c} y^{ - 1} \ln ^A x \leqslant \left| \alpha \right| \leqslant 0.5$$ , c > 2 and ∥c∥ ≥ δ, A is a fixed positive number, and $$\delta = \delta \left( {x,c,A} \right) = \left( {2^{\left[ c \right] + 1} - 1} \right)\left( {A + 2.5} \right) \cdot \frac{{\ln \ln x}} {{\ln x}}$$ .