On the distribution function of the remainder term on bounded remainder sets
A. A. Zhukovaa, A. V. Shutovb
a Russian Academy of National Economy and Public Administration under the President of the Russian Federation (Vladimir Branch)
b Vladimir State University
Bounded remainder sets are sets with bounded by constant
independent of the number of points remainder term of the
multidimensional problem of the distribution of linear function
fractional parts. These sets were introduced by Hecke and studied
by Erdös, Kesten, Furstenberg, Petersen, Szusz, Liardet and
others. Currently, in one-dimensional case full description of
bounded remainder intervals and exact estimates of the remainder
term on such intervals are known. Also some more precise results
about the remainder term are established. Among these results
there are exact formulaes for maximum, minimum and average value
of the remainder term, description of the remainder term as
piecewise linear function, non-monotonic estimates for the
remainder term, estimates of speed of attainment of the remainder
term exact boundaries, etc …In the higher dimensional cases only several examples of bounded
remainder sets are known. Particularly, in recent years V. G.
Zhuravlev, A. V. Shutov, and A. A. Abrosimova introduce a new
construction of some families of multidimensional bounded
remainder sets based on exchanged toric tilings. For introduced
sets we are able not only to prove the boundness of the remainder
term but to compute exact values of its minimum, maximum, and
average. In the present work we study more subtle property of the
remainder term on bounded remainder sets based on exchanged toric
tilings: its distribution function.
It is proved that the remainder term is uniformly distributed only
in one-dimensional case. An algorithm for computation of the
normalized distribution function is given. Some structural results
about this function are proved. For some two-dimensional families
of bounded remainder sets their normalized distribution functions
are clealy calculated.
Bibliography: 31 titles.
distribution modulo one, bounded remainder sets, exchanged toric tilings, distribution function.
PDF file (962 kB)
A. A. Zhukova, A. V. Shutov, “On the distribution function of the remainder term on bounded remainder sets”, Chebyshevskii Sb., 17:1 (2016), 90–107
Citation in format AMSBIB
\by A.~A.~Zhukova, A.~V.~Shutov
\paper On the distribution function of the remainder term on bounded remainder sets
\jour Chebyshevskii Sb.
Citing articles on Google Scholar:
Related articles on Google Scholar:
|Number of views:|