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Hecke’s theorem on the distribution of fractional parts on the unit circle is generalized to the tori TD = RD/L of arbitrary dimension D. It is proved that |δk(i)| ≤ ck n for i = 0, 1, 2, . . . , where δk(i) = rk(i) − iak is the deviation of the number rk(i) of returns in i steps into T D k ⊂ TD for the points of an Sβ-orbit from its mean value ak = vol(T D k )/vol(T D), where vol(Tk ) and vol(T D) denote the volumes of the tile Tk and of the torus T D. The tiles Tk in question have the following property: for the torus TD there exists a development TD ⊂ RD such that a shift Sα of the torus TD is equivalent to some exchange transformation of the corresponding tiles TD k in a partition of the development T D = TD 0 TD 1 · · · TD D . The torus shift vectors Sα, Sβ satisfy the condition α ≡ nβ mod L, where n is any natural number, and the constants ck in the inequalities are expressed in terms of the diameter of the development TD. Introduction Let α be a real number, and let β = 1 n (α + b), where n = 1, 2, 3, . . . and b is an arbitrary integer. We define counting functions by r0(i) = {j; {jβ} < 1− α, 0 ≤ j < i}, r1(i) = {j; {jβ} ≥ 1− α, 0 ≤ j < i}, (0.1) where {x} denotes the fractional part of a number x. Consider the deviations δ0(i) = r0(i)− i(1− α), δ1(i) = r1(i)− iα of r0(i) and r1(i) from their mean values, where 1 − α and α are the lengths of the intervals within which the fractional parts {jβ} fall. Hecke proved the following theorem concerning δ0(i), δ1(i). Theorem 1 (Hecke [3]). Suppose α is irrational. Then the inequalities (0.2) |δ0(i)| ≤ n, |δ1(i)| ≤ n hold true for i = 0, 1, 2, . . . Our goal is to generalize the Hecke theorem to the tori T of arbitrary dimension D. First, observe the following analogy. The functions r0(i) and r1(i) can be related to rotations of the circle. Let (0.3) Sα : x → x+ α mod 1 2010 Mathematics Subject Classification. Primary 11K60; secondary 11H06.

Multidimensional Hecke theorem on the distribution of fractional parts