Geodesic
Short Weyl sums and their applications
Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 232-247

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We shall study the behavior of short Weyl sums of the form $$ T(\alpha ,x,y)=\sum_{x-y\leq x}e(\alpha m^n) $$ on major arcs and obtain an asymptotic formula for the number of representations of a sufficiently large positive integer $N$ as a sum of 33 fifth powers of positive integers $x_i$, that satisfy $ \left|x_i-\left(\dfrac{N}{33}\right)^{\frac 15}\right|\le H$, $H\ge N^{\frac 15-\frac{1}{340}+\varepsilon}$. Bibliography: 17 titles.
Keywords: Short Weyl sums, Almost equal summands, Circle metods, Waring's problem. 
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Z. Kh. Rakhmonov; N. N. Nazrubloev; A. O. Rakhimov. Short Weyl sums and their applications. Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 232-247. http://geodesic.mathdoc.fr/item/CHEB_2015_16_1_a12/