Geodesic
Trigonometric Sums over One-Dimensional Quasilattices of Arbitrary Codimension
Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 781-793

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A new class of one-dimensional quasilattices parametrized by the translations of the torus is introduced. For this class, parameter-dependent trigonometric sums over points of quasilattice are considered. Nontrivial estimates of the trigonometric sums under consideration are obtained. For a number of trigonometric sums of special form, asymptotic formulas are derived. It is proved that the distribution of points of quasilattices is uniform modulo $h$ for almost all $h$. Earlier similar results were obtained in the particular case of quasilattices parametrized by the rotations of the circle.
Keywords: trigonometric sum, codimension, bounded remainder set, tiling of the torus, Weyl's uniform distribution theorem, averaged lattice value, Koksma–Hlawka inequality
Mots-clés : quasilattice, orbit structure. 
@article{MZM_2015_97_5_a11,
     author = {A. V. Shutov},
     title = {Trigonometric {Sums} over {One-Dimensional} {Quasilattices} of {Arbitrary} {Codimension}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {781--793},
     publisher = {mathdoc},
     volume = {97},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_97_5_a11/}
}
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A. V. Shutov. Trigonometric Sums over One-Dimensional Quasilattices of Arbitrary Codimension. Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 781-793. http://geodesic.mathdoc.fr/item/MZM_2015_97_5_a11/