Waring’s problem involving natural numbers of a special type

In 2008–2011, we solved several well–known additive problems such that Ternary Goldbach's Problem, Hua Loo Keng's Problem, Lagrange's Problem with restriction on the set of variables. Asymptotic formulas were obtained for these problems. 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.
Suppose that $k\ge 2$ and $n\ge 1$ are naturals. Consider the equation
$$ \qquad\qquad\qquad\qquad\qquad\qquad x_1^n+x_2^n+\ldots+x_k^n=N\qquad\qquad\qquad\qquad\qquad\qquad\qquad(1) $$
in natural numbers $x_1, x_2, \ldots, x_k$. The question on the number of solutions of the equation (1) is Waring's problem. Let $\eta$ be the irrational algebraic number, $n\ge 3$,
$$k\ge k_0 = \left\{
\begin{array}{ll} 2^n+1, & \hbox{if $3\le n\le 10$,}\\ 2[n^2(2\log n+\log \log n +5)], &\hbox{if $n>10$}. \end{array}
\right.$$
In this report we represent the variant of Waring's Problem involving natural numbers such that $a\le\{\eta x_i^n\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $[0,1)$.
Let $J(N)$ be the number of solutions of (1) in natural numbers of a special type, and $I(N)$ be the number of solutions of (1) in arbitrary natural numbers. Then the equality holds
$$J(N)\sim I(N)\sigma_k(N,a,b).$$

The series $\sigma_k(N,a,b)$ is presented in the main term of the asymptotic formula in this problem as well as in Goldbach's Problem, Hua Loo Keng's Problem.
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