Short Sums with a Noninteger Power of a Natural Number

We establish a nontrivial estimate for a short trigonometric sum of the form $\sum_{x-y<n\le x}e(\alpha [n^c])$, where $y\ge \sqrt{2cx}\,{\mathscr L}^A$, $A\ge 1$ is a fixed number, ${\mathscr L}=\ln x$, and $c$ is a noninteger satisfying the conditions
$$ 1<c\le \log_2{\mathscr L}-\log_2 \ln {\mathscr L}^{6A},\qquad \|c\|\ge(2^{[c]+1}-1)(A+1){\mathscr L}^{-1}\ln{\mathscr L}. $$