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. 
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     author = {Z. Kh. Rakhmonov},
     title = {Generalization of {Waring's} problem for nine almost proportional cubes},
     journal = {\v{C}eby\v{s}evskij sbornik},
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     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_3_a4/}
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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/