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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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