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Multidimensional Hecke theorem on the distribution of fractional parts. (English. Russian original) Zbl 1273.11121
St. Petersbg. Math. J. 24, No. 1, 71-97 (2013); translation from Algebra Anal. 24, No. 1, 95-130 (2012).
Summary: Hecke’s theorem on the distribution of fractional parts on the unit circle is generalized to the tori $$\mathbb T^D=\mathbb R^D/L$$ of arbitrary dimension $$D$$. It is proved that $$|\delta _k(i)|\leq c_kn$$ for $$i=0,1,2,\dots$$, where $$\delta _k(i)=r_k(i)-ia_k$$ is the deviation of the number $$r_k(i)$$ of returns in $$i$$ steps into $$\mathbb T_k^D\subset\mathbb T^D$$ for the points of an $$S_\beta$$-orbit from its mean value $$a_k=\mathrm{vol}(\mathbb T_k^D)/\mathrm{vol}(\mathbb T^D)$$, where $$\mathrm {vol}(\mathbb T_k^D)$$ and $$\mathrm{vol}(\mathbb T^D)$$ denote the volumes of the tile $$\mathbb T_k^D$$ and of the torus $$\mathbb T^D$$. The tiles $$\mathbb T_k^D$$ in question have the following property: for the torus $$\mathbb T^D$$ there exists a development $$T^D\subset\mathbb R^D$$ such that a shift $$S_\alpha$$ of the torus $$\mathbb T^D$$ is equivalent to some exchange transformation of the corresponding tiles $$T_k^D$$ in a partition of the development $$T^D= T_0^D \sqcup T_1^D\sqcup\dots\sqcup T_D^D$$. The torus shift vectors $$S_\alpha$$, $$S_\beta$$ satisfy the condition $$\alpha\equiv n\beta\bmod L$$, where $$n$$ is any natural number, and the constants $$c_k$$ in the inequalities are expressed in terms of the diameter of the development $$T^D$$.

##### MSC:
 11K60 Diophantine approximation in probabilistic number theory 11J54 Small fractional parts of polynomials and generalizations 11K38 Irregularities of distribution, discrepancy
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