On normal lattice configurations and simultaneously normal numbers
Set \(F_{\mathbb{N}} = F_{N_1,\ldots,N_d} = \{ (f_1,\ldots,f_d)\in {\mathbb{N}}^d: 0\leq f_j < N_j\;(1\leq j\leq d)\}\). A \(d\)-dimensional lattice configuration \(\omega: {\mathbb{N}}^d \to \Delta = \{0,1,\ldots, q-1\}\) is called normal if every pattern \(G_{\mathbf h} = \{ g_{\mathbf i} = g_{i_1,\ldots,i_d} \in \Delta : 0\leq i_j < h_j\;(1\leq j\leq d) \}\) (with positive integers \(h_1,\ldots, h_d\)) is attained with the same asymptotic frequency: \[ \# \{ {\mathbf n} \in F_{\mathbb{N}}: \omega({\mathbf n} + {\mathbf i}) = g_{\mathbf i},\;({\mathbf i}\in F_{\mathbf h} \} = N_1\cdots N_d q^{-h_1\cdots h_d} + o(N_1\cdots N_d), \] where \(\max(N_1,\ldots, N_d)\to\infty\). This generalizes the notion of sequences which are normal to base \(q\). The first observation is that a two-dimensional lattice configuration \(\omega(m,n)\) is normal if and only if for every \(s\geq 1\) the double sequence \[ (\{\alpha_m q^n\},\ldots, \{\alpha_{m+s-1} q^n\})_{m,n\geq 1} \] is uniformly distributed in \([0,1)^s\), where \[ \alpha_m = \sum_{n=1}^\infty \omega(m,n) q^{-n}. \] Thus, the behaviour of the (\(s\)-dimensional) discrepancy \(D_{N,M}\) of the sequence \[ (\{\alpha_m q^n\},\ldots, \{\alpha_{m+s-1} q^n\})_{m\leq M,n\leq N} \] may be considered as a quantitative measure for \(\omega\) being normal. The main purpose of the paper is now to construct (more or less explicitly) numbers \(\alpha_m\) of the above form such that the discrepancy is bounded by \[ D_{M,N} = O((MN)^{-1}(\log MN)^{2s+4}\log^2 \log (MN)) \] (and a similar construction if one uses different \(q\)-s). This bound is much better than previously known ones and optimal despite logarithmic factors. The proof is based on Erdős-Turán-Koksma's inequality and is very technical and involved.
scientific article; zbMATH DE number 1770783