On the distribution function of the remainder term on bounded remainder sets

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.
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