Geodesic
On the distribution of points on the modular hyperbola
Dalʹnevostočnyj matematičeskij žurnal, Tome 14 (2014) no. 2, pp. 141-155

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It is known that solutions of determinant equation $\bigl|\begin{smallmatrix} a x\\ y z \end{smallmatrix}\bigr|=q$ are uniformly distributed. The article contains similar results when the variables satisfy additional restrictions $(a,x)=1$ or $(a,x,y,z)=1$. 
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     author = {V. I. Bernik and A. V. Ustinov},
     title = {On the distribution of points on the modular hyperbola},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {141--155},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2014_14_2_a1/}
}
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V. I. Bernik; A. V. Ustinov. On the distribution of points on the modular hyperbola. Dalʹnevostočnyj matematičeskij žurnal, Tome 14 (2014) no. 2, pp. 141-155. http://geodesic.mathdoc.fr/item/DVMG_2014_14_2_a1/