Means over orbits of multidimensional lattices
Let \(p_1, \dots, p_s\in\mathbb{N} \backslash\{1\}\) and \(P_i=p_i^{h_i}\) with \(h_i \in \mathbb{N}\), \(i=1, \dots, s\). For an arbitrary \(s\)-dimensional lattice \(X=\{(x_{k1}, \dots, x_{ks}),\;k=0, \dots, N-1\}\) in the unit cube \(G_s=[0,1)^s\), \(s\geq 1\), the author constructs a group of arithmetic shifts of \(G_s\) \[ G(\vec P)\simeq (\mathbb{Z}/P_1 \mathbb{Z}) \times \cdots \times (\mathbb{Z}/P_s \mathbb{Z}) \] and a group of digital shifts of \(G_s\) \[ G^*(\vec P) \simeq (\mathbb{Z}/p_1 \mathbb{Z})^{h_1} \times \cdots \times (\mathbb{Z}/p_s \mathbb{Z})^{h_s}. \] Let \(X(\vec t)\) (resp. \(XC(\vec t))\) be the image of \(X\) under the transformation \(g(\vec t) \in G(\vec P)\) (resp. \(g^*(\vec t) \in G^*(\vec P))\), and \(G(\vec P)X\) (resp. \(G^*(\vec P)X)\) be the corresponding orbit. For a given weight \(\vec \rho = (\rho,\rho_0, \dots, \rho_{N-1})\), define the local deviation of \(X\) as \[ D(X,\vec \alpha, \vec \rho)=\sum^{N-1}_{k=0} \rho_k \prod^s_{i=1} \chi(x_{ki}, \alpha_i)-\rho N\alpha_1, \dots,\alpha_s, \] where \(\chi (x,\alpha)\) is the characteristic function of the interval \([0,\alpha)\), and the \(q\)-deviation of \(X\) as \[ D_{s,q} (X, \vec\rho) = \int^1_0 \cdots \int^1_0 \bigl|D(X,\vec \alpha, \vec\rho) \bigr|^q d \vec\alpha, \quad q\geq 0. \] We also denote the arithmetic means of \(q\)-deviation over orbits by \[ \sigma_{s,q} \bigl(G(\vec P)X \bigr)=(P_1 \dots P_s)^{-1} \sum_{g (\vec t) \in G(\vec P)} D_{s,q} \bigl (X(\vec t), \vec\rho \bigr) \] and \[ \sigma_{s,q} \bigl(G^*(P)X\bigr) = (P_1\dots P_s)^{-1}\sum_{g^*(\vec t)\in G^*(\vec p)}D_{s,q} \bigl(XC (\vec t), \vec\rho \bigr). \] The author's main result is that \(\sigma_{s,q} (G(\vec P)X) = \sigma_{s,q} (G^* (\vec P) X)\) for any lattice \(X\) and \(q\geq 0\).
arithmetic means of \(q\)-deviation over orbits
scientific article; zbMATH DE number 898910