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

TL;DR

This paper provides discrepancy bounds for $eta$-adic Halton sequences where each $eta_i$ is an $m$-bonacci number, using dynamical systems and geometric properties of Rauzy fractals to relate sequences to rotations on high-dimensional tori.

Contribution

It introduces new discrepancy estimates for $eta$-adic Halton sequences with $m$-bonacci components, employing novel dynamical and geometric techniques.

Findings

Discrepancy bounds are established for sequences with $m$-bonacci $eta_i$.

Sequences are related to rotations on high-dimensional tori.

Classical methods estimate discrepancies via Schmidt's Subspace Theorem.

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 $\boldsymbol{\beta}$-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 $\beta_i$ 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.