Short exponential sums of Weyl's type

The talk is devoted to the investigation of short Weyl's exponential sums of the type
$$ T(\alpha;x,y)\,=\,\sum_{x-y<m\leq x}e(\alpha m^n),\quad \alpha\,=\,\frac{a}{q}\,+\,\lambda , \quad (a,q)=1, \quad q\leq \tau ,\quad |\lambda|\,\leq\, \frac{1}{q\tau}, $$
and theirs applications to the problem of distribution of fraction parts of the polinomial with the argument from a short segment, and to some additive problems with almost equal summands.