A metric result for special sequences related to the Halton sequences
The Halton sequence \((\mathbf{y}_n)_{n\geq 0}\) is a well-known and important example of a low discrepancy sequence. It is defined via the so-called radical inverse function \(\phi_b\) which mirrors the \(b\)-adic expansion of an integer \(n\) at the decimal point and gives a number in the unit interval \([0,1)\). The Halton sequence in pairwise coprime integer bases \(b_1, \ldots, b_d \geq 2\) is a \(d\)-dimensional sequence of points in \([0,1)^d\) defined by \[ \mathbf{y}_n = \left( \phi_{b_1}(n), \ldots, \phi_{b_d}(n)\right). \] The author of the paper under review studies related sequences of the form \[ \mathbf{x}_n = \left( \phi_{b_1}(\lfloor n \beta \rfloor ), \ldots, \phi_{b_d}(\lfloor n \beta \rfloor)\right), \] for nonzero \(\beta \in \mathbb{R}\) and pairwise coprime integers \(b_1, \ldots, b_d \geq 2\). The main result of the paper (Theorem 1) provides an upper bound on the discrepancy of such sequences for almost all \(\beta \in \mathbb{R}\) (in the sense of Lebesgue measure) which is close to the upper bound satisfied by low discrepancy sequences.
scientific article; zbMATH DE number 6810591