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Short Weyl sums and their applications. (Russian. English summary) Zbl 1437.11122

Summary: We shall study the behavior of short Weyl sums of the form \[ T(\alpha ,x,y)=\sum_{x-y<m\leq x}e(\alpha m^n) \] 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 \( \left|x_i-\left(\dfrac{N}{33}\right)^{\frac 15}\right|\le H, H\ge N^{\frac 15-\frac{1}{340}+\varepsilon}\).

MSC:

11L07 Estimates on exponential sums
11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
11L03 Trigonometric and exponential sums (general theory)
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References:

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