Asymptotic formula in the Waring's problem with almost proportional summands

Z. Kh. Rakhmonov ; F. Z. Rahmonov

Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 139-168

Résumé

For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1, r}$, satisfying the conditions $$ |x_i^n-\mu_iN|\le H, H\ge N^{1-\theta(n,r)+\varepsilon}, \theta(n,r)=\frac2{(r+1)(n^2-n)}, $$ where $\mu_1, \ldots, \mu_r$ are positive fixed numbers, and $\mu_1 + \ldots + \mu_n = 1$. This result strengthens the theorem of E.M. Wright.

Détail

Keywords: Waring problem, almost proportional summands, short exponential sum of G. Weyl, small neighborhood of centers of major arcs.