Multidimensional Hecke theorem on the distribution of fractional parts

V. G. Zhuravlev
2012 St. Petersburg Mathematical Journal  
Hecke's theorem on the distribution of fractional parts on the unit circle is generalized to the tori where vol(T D k ) and vol(T D ) denote the volumes of the tile T D k and of the torus T D . The tiles T D k in question have the following property: for the torus T D there exists a development T D ⊂ R D such that a shift S α of the torus T D is equivalent to some exchange transformation of the corresponding tiles T D k in a partition of the development T D = T D 0 T D 1 · · · T D D . The torus
more » ... shift vectors S α , S β satisfy the condition α ≡ nβ mod L, where n is any natural number, and the constants c k in the inequalities are expressed in terms of the diameter of the development T D .
doi:10.1090/s1061-0022-2012-01232-x fatcat:wykz4z6tqjgulpepki7mqo7hnm