On corner avoidance of \(\beta\)-adic Halton sequences (Q292655)

The authors show that the extended \(s\)-dimensional \({\boldsymbol \beta}\)-adic Halton sequence (extension of \(\boldsymbol \beta\)-adic Halton sequence given in [the first author et al., Ergodic Theory Dyn. Syst. 35, No. 3, 895--909 (2015; Zbl 1395.11106)]) is uniformly distributed in \([0,1)^s\). Let \({\mathbf h}=(h^{(1)},\ldots, h^{(s)})\in \{0,1\}^s\) be a corner of the unit cube \([0,1)^s\). The hyperbolic distance for \({\mathbf x}=(x^{(1)},\ldots, x^{(s)})\in [0,1)^s\) to the corner \({\mathbf h}\) is defined by \[ \|{\mathbf x}\|_{\mathbf h}=\prod_{i=0}^s |x^{(i)}-h^{(i)}|. \] Using Schmidt's subspace theorem, they prove that the \(s\)-dimensional \(\boldsymbol \beta\)-adic Halton sequence \((\psi_{\boldsymbol \beta}(n))_{n>0}\) avoids the corners: for any \(\varepsilon>0\) there exists \(C_{\varepsilon, {\boldsymbol \beta}}>0\) such that \[ \|\psi_{\boldsymbol \beta}(N)\|_{\mathbf h}>\frac{C_{\varepsilon, \boldsymbol \beta}}{N^{H/2+\varepsilon}}, \] where \(H=2\) if \({\mathbf h}=(0,\ldots, 0)\); \(H=s\) if \({\mathbf h}=(1,\ldots 1)\); \(H=1+\sum_{i=1}^s h^{(i)}\) otherwise.