Regularities of the distribution of beta-adic van der Corput sequences
Résumé
For Pisot numbers $\beta$ with irreducible $\beta$-polynomial, we prove that the discrepancy function $D(N,[0,y))$ of the $\beta$-adic van der Corput sequence is bounded if and only if the $\beta$-expansion of $y$ is finite or its tail is the same as that of the expansion of 1. If $\beta$ is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length $y\not\in\mathbb Q(\beta)$. We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse $\beta$-substitution.
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