Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -adic Halton Sequences

Abstract

Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost 20 years ago Ninomiya defined analogues of van der Corput sequences for β-numeration and proved that they also form low-discrepancy sequences if β is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that \(\boldsymbol{\beta }\)-adic Halton sequences are equidistributed for certain parameters \(\boldsymbol{\beta }= (\beta _{1},\ldots,\beta _{s})\) using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for \(\boldsymbol{\beta }\)-adic Halton sequences for which the components β i are m-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate \(\boldsymbol{\beta }\)-adic Halton sequences to rotations on high dimensional tori. The discrepancies of these rotations can then be estimated by classical methods relying on W.M. Schmidt’s Subspace Theorem.