Geodesic
Multidimensional generalization of sums of fraction parts and their applications to number theory
Čebyševskij sbornik, Tome 14 (2013) no. 1, pp. 104-118

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In the paper a new multidimensional generalization of fraction part function is introduced. We obtain a formula which express the number of points from the orbit of irrational shift on multidimensional torus, lying in a given domain, in the terms of sums of multidimensional fraction parts. Also we give some application of this formula to various number-theoretic problems. 
@article{CHEB_2013_14_1_a10,
     author = {A. V. Shutov},
     title = {Multidimensional generalization of sums of fraction parts and their applications to number theory},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {104--118},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2013_14_1_a10/}
}
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A. V. Shutov. Multidimensional generalization of sums of fraction parts and their applications to number theory. Čebyševskij sbornik, Tome 14 (2013) no. 1, pp. 104-118. http://geodesic.mathdoc.fr/item/CHEB_2013_14_1_a10/