On the solvability of Waring's equation involving natural numbers of a special type

This paper is a continuation of our research on additive problems of number theory with variables that belong to some special set. We have solved several well-known additive problems such that Ternary Goldbach's Problem, Hua Loo Keng's Problem, Lagrange's Problem, Waring's Problem. Asymptotic formulas were obtained for these problems with restriction on the set of variables. The main terms of our formulas differ from ones of the corresponding classical problems. In the main terms the series of the form
$$ \sigma_k (N,a,b)=\sum_{|m|<\infty} e^{2\pi i m(\eta N-0,5 k(a+b))} \frac{\sin^k \pi m (b-a)}{\pi ^k m^k}. $$
appear. These series were investigated by the authors.
Let $\eta$ be the irrational algebraic number, $a$ and $b$ are arbitrary real numbers of the interval $[0,1]$. There are natural numbers $x_1, x_2, \ldots, x_k$ such that
$$a\le\{\eta x_i^n\}<b.$$
In this paper we evaluate the smallest $k$ for which the equation
$$ x_1^n+x_2^n+\ldots+x_k^n=N $$
is solvable.
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