On variants of the Halton sequence (Q303851)

This paper deals with generalizations of a classical example of uniformly distributed sequences in the \(s\)-dimensional unit cube \([0,1)^s\), namely Halton sequences. The classical Halton sequence in basis \(\beta_1,\ldots,\beta_s\), where \(\beta_1,\ldots,\beta_s\) are integers greater than or equal to two, is defined by applying the radical inverse function \(\phi_{\beta_i}\) \(1\leq i\leq s\), to the nonnegative integers, thus yielding a sequence \(\left((\phi_{\beta_1}(n),\ldots,\phi_{\beta_s}(n) )\right)_{n\geq 0}\). This definition can be generalized to real numbers \(\beta_1,\ldots,\beta_s\) strictly greater than one, by making use of suitable enumeration systems. The question addressed in this article is under which circumstances the sequence \(((\phi_{\beta_1}(k_n),\ldots,\phi_{\beta_s}(k_n) ))_{n\geq 0}\) is uniformly distributed in \([0,1)^s\), for real \(\beta_1,\ldots,\beta_s\) and a sequence \((k_n)_{n\geq 1}\) of positive integers. The main result in the paper shows that, for suitable enumeration systems, uniform distribution is guaranteed if the sequence \((k_n)_{n\geq 0}\) is Hartman uniformly distributed and good universal.