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.
This publication is cited in the following 3 articles:
A. V. Shutov, “O chislakh s zadannymi poslednimi tsiframi razlozheniya po lineinoi rekurrentnoi posledovatelnosti”, Dalnevost. matem. zhurn., 24:1 (2024), 141–150
A. V. Shutov, “Podstanovki i mnozhestva ogranichennogo ostatka”, Chebyshevskii sb., 19:2 (2018), 501–522
A. V. Shutov, “Trigonometric Integrals over One-Dimensional Quasilattices of Arbitrary Codimension”, Math. Notes, 99:4 (2016), 590–597