On variants of the Halton sequence

A generalisation of the classical Halton sequence ((phi _{beta }(n))_{nin 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))_{jin mathbb {N}}) for the sequence of integers ((k_j)_{jge 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.