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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. 67, N 6, 2012.].

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.