Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -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 βnumeration 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 β-adic Halton sequences are equidistributed for certain parameters β = (β 1 , . . . , βs) using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for β-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 β-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.∞ j=0 ε j (n)G j ,

The level of distribution of the sum-of-digits function of linear recurrence number systems

“…Finally, we note that, starting with Barat and Grabner [2], van der Corput and Halton type sequences using linear recurrence bases are investigated. Work on this topic can be found in Ninomiya [24], Hofer et al [20], and Thuswaldner [33].…”

The level of distribution of the sum-of-digits function of linear recurrence number systems

“…Finally, we note that, starting with Barat and Grabner [3], van der Corput and Halton type sequences using linear recurrence bases are investigated. Work on this topic can be found in Ninomiya [24], Hofer et al [20], and Thuswaldner [33].…”