Wolfgang Steiner: The local discrepancy of generalized van der Corput sequences

This is a joint work with Klaus Scheicher.

Let $\beta>1$ be a real number and

$$S=\left\{(\ldots,\epsilon_2,\epsilon_1)\in\{0,1,\ldots,\lfloor\be... ...}\beta+\frac{\epsilon_{j+1}}{\beta^2}+\cdots<1 \mbox{ for all } j\ge1\right\}$$

be the set of all (inversed) greedy expansions in base $\beta$. Then the $n$-th element of the generalised van der Corput sequence in base $\beta$ is defined by the $n$-th element of $S$ with respect to the lexicographical order, $x_n=\epsilon_1/\beta+\epsilon_2/\beta^2+\cdots$. If the expansion of 1 in base $\beta$ is finite or ultimately periodic, then we have a substitution such that the digits of $x_n$ are the lengths of the words in the expansion of $n$ with respect to this substitution. This allows us to get explicit formulae for the local discrepancy ( $\char93 \{n<N: x_n\in[0,y)\}-Ny$) of the sequence. If $\beta$ is a Pisot number and the $\beta$-polynomial is the minimal polynomial of $\beta$, then the local discrepancy is bounded (in $N$) if and only if either the $\beta$-expansion of $y$ is finite or its tail is the same as the tail of the expansion of 1.

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