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

An estimate for short exponential sums
$$S_c(\alpha ;x,y)=\sum_{x-y<n\le x}e(\alpha [n^c])$$
is obtained for $y\ge x^{\frac{1}{2}}\ln^A x$, $x^{1-c}y^{-1}\ln^Ax\le|\alpha|\le 0,5$, $c>2$ and $\|c\|\ge\delta$ where $A$ is a fixed positive number and $\delta=\delta (x,c,A)=\left(2^{[c]+1}-1\right)(A+2,5)\cdot\frac{\ln\ln x}{\ln x}$.