On corner avoidance of $\mathbf{\beta }$-adic Halton sequences

Abstract

We consider the corner avoiding property of $s$-dimensional $\mathbf{\beta }$-adic Halton sequences. After extending this class of point sequences in an intuitive way, we show that the hyperbolic distance between each element of the sequence and the closest corner of $[0,1{)}^s$ is $\mathcal{O}\left(\frac{1}{N^{s/2+\epsilon }}\right)$, where $N$ denotes the index of the element. In our proof we use tools from Diophantine analysis, more precisely, we apply Schmidt's Subspace Theorem.