
Convex rhombic dodecahedron and parametric BRsets
A. A. Osipova^{} ^{} Russian University of Cooperation, Vladimir Branch
Abstract:
The paper is devoted to the important problem of number theory:
bounded remainder sets.
We consider the point orbits on lowdimensional tori. Any starting
point generates the orbit under an irrational shift of the torus.
The orbit is everywhere dense and uniformly distributed on the
torus if the translation vector is irrational. Denote by $r(i)$ a
function that gives the number of the orbit points which get some
domain $T$. Then we have the formula $ r(i) = i \: \mathrm{ vol}
(T) + \delta(i)$, where $\delta(i)=o(i)$ is the remainder. If the
boundaries of the remainder are limited by a constant, then $T$ is
a bounded remainder set (BRset).
The article introduces a new BRsets construction method, it is
based on tilings parametric polyhedra. Ñonsidered polyhedra are
the torus development. Torus development should be to tile into
figures, that can be exchanged, and we again obtain our torus
development. This figures exchange equivalent shift of the
torus.
Author have constructed tillings with this property and
twodimensional BRsets. The considered method gives exact
estimates and the average value of the remainder. Also we obtain
the optimal BRsets which have minimal values of the remainder.
These BRsets generate the strong balanced words (a
multidimensional analogue of the
Sturmian words).
The above method is applied to the case of threedimensional torus
in this paper. Also we obtain exact estimates and the average
value of the remainder for constructed sets.
Bibliography: 22 titles.
Keywords:
bounded remainder sets, distribution of fractional parts, toric development.
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UDC:
511.34 Received: 12.12.2015 Accepted:11.03.2016
Citation:
A. A. Osipova, “Convex rhombic dodecahedron and parametric BRsets”, Chebyshevskii Sb., 17:1 (2016), 160–170
Citation in format AMSBIB
\Bibitem{Osi16}
\by A.~A.~Osipova
\paper Convex rhombic dodecahedron and parametric BRsets
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 160170
\mathnet{http://mi.mathnet.ru/cheb461}
\elib{https://elibrary.ru/item.asp?id=25795079}
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http://mi.mathnet.ru/eng/cheb461 http://mi.mathnet.ru/eng/cheb/v17/i1/p160
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