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 Mat. Sb. (N.S.), 1970, Volume 82(124), Number 3(7), Pages 360–370 (Mi msb3455)  An asymptotic formula for the number of solutions of a Diophantine equation

M. I. Israilov

Abstract: Suppose that $k$, $s$, $m_1,…,m_k$, $m_1',…,m_s'$ are fixed positive integers, $m$ is a fixed integer, $p$ is an increasing positive integer, and suppose that a sequence of integers $\{n_k\}$ satisfies the following conditions: 1) $n_{k+1}\geqslant n_k(1+k^{-1/2+\varepsilon})$ , where $\varepsilon>0$ is arbitrarily small; 2) for fixed $m,n,a,B$, the number of solutions of the Diophantine equation
$$mn_{x+a}-nn_x=B$$
in $x$ in the half-open interval $[0,p)$ does not exceed some constant $q$ which does not depend on $m,n,a,B$.
Under these assumptions, an asymptotic formula with remainder term is derived for the number of solutions of the Diophantine equation
$$m_1n_{x_1}+…+m_kn_{x_k}=m_1'n_{y_1}+…+m_s'n_{y_s}+m$$
in integers $0\leqslant x_1,…,x_k$; $y_1,…,y_s<p$.
The results obtained extend and refine several results obtained by other authors.
Bibliography: 7 titles. Full text: PDF file (838 kB) References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1970, 11:3, 327–338 Bibliographic databases:  UDC: 511.222
MSC: 11D75, 26B35, 11Bxx

Citation: M. I. Israilov, “An asymptotic formula for the number of solutions of a Diophantine equation”, Mat. Sb. (N.S.), 82(124):3(7) (1970), 360–370; Math. USSR-Sb., 11:3 (1970), 327–338 Citation in format AMSBIB
\Bibitem{Isr70} \by M.~I.~Israilov \paper An~asymptotic formula for the number of solutions of a~Diophantine equation \jour Mat. Sb. (N.S.) \yr 1970 \vol 82(124) \issue 3(7) \pages 360--370 \mathnet{http://mi.mathnet.ru/msb3455} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=323714} \zmath{https://zbmath.org/?q=an:0211.37503|0215.34603} \transl \jour Math. USSR-Sb. \yr 1970 \vol 11 \issue 3 \pages 327--338 \crossref{https://doi.org/10.1070/SM1970v011n03ABEH002072} 

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This publication is cited in the following articles:
1. Zeév Rudnick, Alexandru Zaharescu, “The distribution of spacings between fractional parts of lacunary sequences”, form, 14:5 (2002), 691   •   Contact us: math-net2019_06 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019