This article is cited in 2 papers

Generalization of Waring's problem for nine almost proportional cubes

Z. Kh. Rakhmonov

A. Dzhuraev Institute of Mathematics (Dushanbe)

Abstract: 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.

UDC: 511.32

Received: 29.03.2023
Accepted: 12.09.2023

DOI: 10.22405/2226-8383-2023-24-3-71-94