Block Additive Functions on Gaussian Integers

Michael Drmota

Technische Universität Wien, Austria

Coauthor(s): Peter Grabner

It is well known that every Gaussian integer $z\in \mathbb{Z}[i]$ can be uniquely represented as

$$z = \sum_{j\ge 0} \varepsilon_j(z) q^j$$   with$$\quad q = -a+i,$$

where the digits $\varepsilon_j(z)$ are from the set $\{0,1,\ldots,\vert q\vert^2-1\}$. In this talk we report on recent results on the distribution of digital blocks $B = (\eta_0,\eta_1,\ldots,\eta_L)$ in digital expansions of that kind.

In particular we prove a central limit theorem for the number of occurrences $a_B(z)$ of a given block in the digital expansion of $z$ (together with asymptotic expansions for the moments and a local limit theorem) if $\vert z\vert^2 < N$ and $N\to\infty$. Further we prove uniform distribution results in residue classes and modulo 1.

The proofs rely on asymptotic expansion of the generating function $\sum_{\vert z\vert^2<N} x^{a_B(z)}$ that can be obtained by applying the inverse Mellin transform on the Dirichlet series $\sum_{z\ne 0} x^{a_B(z)} \vert z\vert^{-2s}$.

This work was supported by the Austrian Science Foundation FWF, projects S9604 and S9605.