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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 37–51 (Mi cheb452)

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

S. A. Gritsenkoabc, N. N. Motkinad

a Financial University under the Government of the Russian Federation, Moscow
b Lomonosov Moscow State University
c Bauman Moscow State Technical University
d National Research University "Belgorod State University"

Abstract: 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.
Bibliography: 23 titles.

Keywords: Waring's Problem, additive problems, numbers of a special type, number of solutions, asymptotic formula, irrational algebraic number.

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UDC: 511.34
Accepted:10.03.2016

Citation: S. A. Gritsenko, N. N. Motkina, “On the solvability of Waring's equation involving natural numbers of a special type”, Chebyshevskii Sb., 17:1 (2016), 37–51

Citation in format AMSBIB
\Bibitem{GriMot16} \by S.~A.~Gritsenko, N.~N.~Motkina \paper On the solvability of Waring's equation involving natural numbers of a special type \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 1 \pages 37--51 \mathnet{http://mi.mathnet.ru/cheb452} \elib{https://elibrary.ru/item.asp?id=25795068}