Waring's problem with almost proportional summands

TL;DR

This paper derives an asymptotic formula for the number of representations of large numbers as sums of almost proportional n-th powers, extending Wright's theorem with new bounds on summand deviations.

Contribution

It introduces a new asymptotic formula for sums of n-th powers with almost proportional summands, improving upon Wright's theorem with refined bounds.

Findings

Derived an asymptotic formula for representations of large N

Extended Wright's theorem with new bounds on summand deviations

Established conditions for summands close to proportionality

Abstract

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 $$ |x_i^n-\mu_iN|\le H,\qquad H\ge N^{1-\theta(n,r)+\varepsilon},\qquad \theta(n,r)=\frac2{(r+1)(n^2-n)}, $$ 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.