Geodesic
Generalization of Waring's problem for nine almost proportional cubes
Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 71-94

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An asymptotic formula is obtained for the number of representations of a sufficiently large natural $N$ as a sum of nine cubes of natural numbers $x_i$, $i=\overline{1,9}$, satisfying the conditions $$ |x_i^3-\mu_iN|\le H, \mu_1+\ldots+\mu_9=1 H\ge N^{1-\frac1{30}+\varepsilon} , $$ where $\mu_1,\ldots,\mu_9$ — positive fixed numbers. This result is a strengthening of E.M.Wright's theorem.
Keywords: Waring's problem, almost proportional Summands, H. Weil's short exponential sum, small neighborhood of centers of major arcs. 
Z. Kh. Rakhmonov. Generalization of Waring's problem for nine almost proportional cubes. Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 71-94. http://geodesic.mathdoc.fr/item/CHEB_2023_24_3_a4/
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