ON THE DISTRIBUTION FUNCTION OF THE REMAINDER TERM ON BOUNDED REMAINDER SETS
https://doi.org/10.22405/2226-8383-2016-17-1-90-107
Abstract
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 Erd¨os, 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.
About the Authors
A. A. Zhukova
Federal State Educational Institution of Higher Education "Russian Academy of National Economy and Public Administration under the President of Russian Federation Vladimir branch
Russian Federation
Russian Federation
Candidate of Physical and Mathematical Sciences, head of postgraduate studies, research and international activities Department; Associate Professor, Department of Information Technology
A. V. Shutov
Federal State Educational Institution of Higher Education "Vladimir State University named after Alexander G. and Nicholay G. Stoletovs"(VlSU)
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Associate Professor of Control and Informatics in Technical and Economic Systems.
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Review
For citations:
Zhukova A.A., Shutov A.V. ON THE DISTRIBUTION FUNCTION OF THE REMAINDER TERM ON BOUNDED REMAINDER SETS. Chebyshevskii Sbornik. 2016;17(1):90-107. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-90-107
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