Short exponential sums with a fractional powers of natural numbers

The talk is devoted to the estimation of the exponential sum of the type
$$ \sum_{x-y<n\le x}\exp(2\pi i\alpha [n^c]), $$
where $y\ge\sqrt{2cx}\,\mathcal{L}^{A}$, $A\ge 1$ is fixed, $\mathcal{L}=\ln x$ and $c$ is non-integer satisfying the conditions
$$ 1<c\leqslant \log_2\mathcal{L}-\log_2\ln(\mathcal{L}^{6A}) , \qquad \|c\|\ge (2^{[c]+1}-1)(A+1) \mathcal{L}^{-1}\ln{\mathcal{L}}. $$