Geodesic
An asymptotic formula for the number of solutions of a Diophantine equation
Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 327-338 
M. I. Israilov. An asymptotic formula for the number of solutions of a Diophantine equation. Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 327-338. http://geodesic.mathdoc.fr/item/SM_1970_11_3_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that $k$, $s$, $m_1,\dots,m_k$, $m_1',\dots,m_s'$ 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 $$ m_1n_{x_1}+\dots+m_kn_{x_k}=m_1'n_{y_1}+\dots+m_s'n_{y_s}+m $$ in integers $0\leqslant x_1,\dots,x_k$; $y_1,\dots,y_s. The results obtained extend and refine several results obtained by other authors. Bibliography: 7 titles.

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[3] M. P. Mineev, “O probleme Tarri dlya bystrodeistvuyuschikh funktsii”, Matem. sb., 46(88) (1964), 451–454 | MR

[4] A. G. Postnikov, Ergodicheskie voprosy teorii sravnenii i teorii diofantovykh priblizhenii, Trudy matem. in-ta AN SSSR im. V. A. Steklova, 82, 1966 | MR

[5] P. Erdos, “On trigonometric sums with gaps”, MTA Math. Kut. Int. Közl., 1962, no. 7, 37–42 | MR

[6] P. X. Mukhutdinov, “Diofantovo uravnenie s matematicheskoi pokazatelnoi funktsiei”, DAN SSSR, 142:1 (1962), 36–38

[7] R. X. Mukhutdinov, “Ob odnoi predelnoi teoreme”, DAN UzSSR, 1965, no. 7