Short sums with a noninteger power of a natural number

We establish a nontrivial estimate for a short trigonometric sum of the form Ωx − y < n ≤ xe(α[nc]), where u ≥ √2cx ℒ6A, A ≥ 1 is a fixed number, ℒ = ln x and c is a noninteger satisfying the conditions $$1 < c \leqslant \log _2 L - \log _2 \ln L^{6A} , \left\| c \right\| \geqslant \left( {2^{\left[ c \right] + 1} - 1} \right)\left( {A + 1} \right)L^{ - 1} \ln L.$$