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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 torusdoi:10.1090/s1061-0022-2012-01232-x fatcat:wykz4z6tqjgulpepki7mqo7hnm