Geodesic
On the distribution of fractional parts of polynomials of two variables
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 222-232

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In the paper, upper bounds for sums of the form $$ \underset{(n_1,n_2)\in\Omega}{\sum\sum}\psi(f(n_1,n_2)), $$ where $\psi(x)=x-[x]-\frac12$, $f(x,y)$ is a polynomial, $(n_1,n_2)\in\mathbb Z^2$, and $\Omega$ is a domain in $\mathbb R^2$, are obtained. One of the upper bounds is of interest, particularly in connection with a lattice point problem considered in Theorem 2. 
@article{ZNSL_2012_404_a13,
     author = {O. M. Fomenko},
     title = {On the distribution of fractional parts of polynomials of two variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {222--232},
     publisher = {mathdoc},
     volume = {404},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a13/}
}
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O. M. Fomenko. On the distribution of fractional parts of polynomials of two variables. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 222-232. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a13/