Let be a real number and
be the set of all (inversed) greedy expansions in base .
Then the -th element of the generalised van der Corput sequence in base
is defined by the -th element of with respect to the
If the expansion of 1 in base is finite or ultimately periodic, then we
have a substitution such that the digits of are the lengths of the words in the
expansion of with respect to this substitution.
This allows us to get explicit formulae for the local discrepancy (
) of the sequence.
If is a Pisot number and the -polynomial is the minimal polynomial of ,
then the local discrepancy is bounded (in ) if and only if either the -expansion
of is finite or its tail is the same as the tail of the expansion of 1.
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Please send comments and corrections to Thomas Klausner.