Short trigonometric sums with not integer degree of natural number / P. Z. Rakhmonov. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2012. № 6. P. 51-55 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].

An estimate for short exponential sums

S_c(α;x,y)=∑_x-y<n≤xe(α[n^c]) << y ln^Ax,
is obtained for y≥x^1/2ln^Ax, x^1-cy^-1ln^Ax≤|α|≤0,5, c>2 and ||c||≥δ where A is a fixed positive number and δ=δ(x,c,A)=(2^[c]+1-1)(A+2,5)(ln ln x)/(ln x).

Key words: short exponential sum, Van der Corput's method, exponential integral, nontrivial estimate.