Discrepancy bounds for $\mathbf{\beta }$-adic Halton sequences

Abstract

Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost twenty years ago Ninomiya defined analogues of van der Corput sequences for $\beta$-numeration and proved that they also form low-discrepancy sequences if $\beta$ is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that $\mathbf{\beta }$-adic Halton sequences are equidistributed for certain parameters $\mathbf{\beta }=({\beta }_1,\dots ,{\beta }_s)$ using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for $\mathbf{\beta }$-adic Halton sequences for which the components ${\beta }_i$ are $m$-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate $\mathbf{\beta }$-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.

arXiv comment15 pages