Multidimensional Hecke theorem on the distribution of fractional parts

Abstract:

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 $|\delta _{k}(i)| \leq c_k n$ 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 $T^D$.