On variants of the Halton sequence

A generalisation of the classical Halton sequence \((\phi _{\beta }(n))_{n\in \mathbb {N}}\) has emerged in recent years based on \(\beta \)-adic expansions of elements of 0, 1). In the case where \(\beta \) is a natural number greater than 1, this reduces to the classical Halton sequence. In this paper, we use ergodic and analytic methods to prove the uniform distribution of a sequence \((\phi _{\beta }(k_j))_{j\in \mathbb {N}}\) for the sequence of integers \((k_j)_{j\ge 0}\), which is both Hartman uniformly distributed and good universal. This builds on earlier work of M. Hofer, M. R. Iaco and R. Tichy in the special case \(k_j =j \ (j=0,1, \ldots )\). Variants of this phenomenon are also studied.