Regularities of the Distribution of Β-adic Van Der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0, y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length y 6∈ Q(β). We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.