Abstract:
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, quasilattice, codimension, bounded remainder set, tiling of the torus, Weyl's uniform distribution theorem, averaged lattice value, Koksma–Hlawka inequality, orbit structure.
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